Essential Poker Math - Alton Hardin

Note: While reading a book whenever I come across something interesting, I highlight it on my Kindle. Later I turn those highlights into a blogpost. It is not a complete summary of the book. These are my notes which I intend to go back to later. Let’s start!

The Two Aspects of Poker

• There are two aspects of analysis in poker. The first is reading your opponents, and the second is using mathematics to ensure we make correct mathematical moves based upon our reads and tells. When we read our opponents, we are gaining reads, tells and tendencies that help us to understand the “range” of hands our opponent can have in his hand. This tells us how likely it is that our opponent has a made hand versus a drawing hand, as well as how strong it is.

• We then use basic poker mathematics to supplement our reads and tells. When we do this, our goal is to ensure that we are maximizing how often we make profitable moves, while minimizing unprofitable ones. In poker, we call profitable moves positive expected value (+EV) plays, and unprofitable ones negative expected value (–EV) plays. Our goal is to make as many +EV plays as possible at the poker table.

• A complete poker player is one that can both read their opponents and use math at the poker table to make mathematically correct +EV plays. A poker player that focuses solely on reading their opponents, neglecting math, is an incomplete poker player. Conversely, a poker player that does not read their opponents, but bases all of their moves on math alone, is also an incomplete poker player. The best and most profitable players in the world are both excellent at reading their opponents and well versed in poker mathematics. As you can see, math in poker, when balanced with a good ability to read your opponents, is essential to your long-term success in the game.

• Exploitative poker: Poker style in which players seek holes in their opponents’ games and exploit them through tells, tendencies, and general weaknesses.

• A hand range is the set of all possible starting hands a hero or villain can have when playing poker.

• Hand range denotion:
  • Any Pocket Pair : 22+ = 22, 33, 44, 55, 66, 77, 88, 99, TT, JJ, QQ, KK, AA
  • Pocket Jacks or Better : JJ+ = JJ, QQ, KK, AA
  • KQ or Better : KQ+ = KQ, AJ, AQ, AK
  • AJ suited or Better: AJs+ = AJs, AQs, AKs

• Poker players do certain things in poker; such as pre-flop raises, isolation raises, squeeze plays, 3-bets, 4-bets, steals, continuation bets, bluffs and so forth, with specific ranges of hands that we can estimate based upon their playing style, tendencies and HUD stats (for online players).

• Understanding and being able to visualize hand ranges is an important skill to have in poker, because how our opponents play provides insight into their possible range of hands. Being able to read our opponents’ range of possible hands is something you should seek to master.

• Effective stack size is the size of the smallest stack between two different players in a hand. This indicates the highest amount of money you can either win or lose in a hand against any one particular opponent.

• You and an opponent are both all-in pre-flop. You have $150 at the start of the hand and your opponent only has $40; therefore, with only $80 in the pot the most either you or your opponent can win or lose at the end of the hand is $40.

• Knowing effective stack sizes is an essential concept in basic poker strategy - how we play a particular hand will vary greatly depending upon our opponents’ stack sizes. As good poker players, we’ll typically have stack sizes of at least 100 big blinds, but our opponents will have stack sizes ranging from 20bb to 400bb. Because some of our opponents will be playing short-stacked and others deep-stacked, we need to take their stack sizes into consideration for every single hand, the reason for this being that people tend to play drastically diverse strategies with different effective stack sizes.

• Stack-to-Pot Ratios – commonly referred to as SPRs – compares the current pot size to your stack size.

• SPR = Effective stack size / Pot size.

• We can think of SPRs as a guide on how committed we are to any particular hand. As a rule of thumb, when SPRs are small, people will tend to be more committed to hands; whereas when SPRs are bigger and stacks are deeper, people will be less committed to hands without the nuts. Another way to look at an SPR is as a “risk-to-reward” ratio, where a person risks his or her effective stack size to win the size of the pot. When effective stack sizes are short, we’re risking less to win the pot, but when effective stack sizes are deep; we are risking a lot to win the pot.

• The main takeaway from SPRs is:
  • Lower SPRs = Smaller Effective Stack Sizes (Short Stackers)
  • Higher SPRs = Larger Effective Stack Sizes (Deep Stackers)
  • Lower SPRs = Commit with Weaker Hands
  • Higher SPRs = Commit with Stronger Hands
• If you’re playing live poker, there are several instant indicators you can use even before playing a single hand of poker to determine if a player is potentially good or bad:
  • Look at their stack sizes. Typically good poker players will have at least a 100bb stack, whereas bad or purely recreational players will often have random-sized short stacks
  • Look at how their chips are stacked. Are they nicely stacked into 20 chip stacks, or erratically into small stacks? If they’re erratic small stacks, they’re probably a bad or recreational player
  • Are they performing chip tricks? Decent regulars will often perform tricks, such as the chip shuffle at the table
  • Are they listening to music? Decent regulars will often also listen to music on their cell phones
  • Are they drinking alcohol or do they appear drunk? If so, they’re probably a bad or recreational player having fun and gambling at the table

• As play commences at the table, take notes on your opponents. If you play live, you’ll be limited to mental notes; however, if you play online you have the ability to write down notes into your poker client or HUD, which is something I highly recommend you do. Look for things outside of the ordinary, as well as telling plays, which will help you categorize your opponents as good or bad. If you are very observant, within one to two orbits of hands you should have a good idea of how your opponents are playing.

• There are three basic types of good poker players:
  • NITs (Really Tight Players)
  • TAGs (Tight Aggressive Players)
  • LAGs (Loose Aggressive Players)

• NITs can be categorized as the scrooges of poker. They are very risk-averse, and only play the very best-of-the-best starting hands pre-flop. Additionally, they will usually only get involved in big pots post-flop with a very strong hand. Most NITs play a very tight and aggressive style of poker, and will play fit-or-fold post-flop. This means that they will only continue with a hand post-flop if they have hit a strong hand or very strong draw. Always be aware of NITs when they are betting or raising; this usually means they have a very strong hand or draw - NITs are not known to bluff.

• Most TAGs are very difficult to play against because they are competent poker players, skilled in all aspects of the game. Unlike most NITs, a TAG is also capable of bluffing in opportune spots. A TAG doesn’t need a made or strong poker hand to bet and be aggressive, which makes them difficult to play against.

• Good loose aggressive opponents – commonly referred to as LAGs – are arguably the toughest type of poker player to play against. The LAG-style of play, when implemented properly, is the most profitable style of poker.

• LAGs are tougher to play against than TAGs, because they play a wider range of hands than TAGs and bluff more often. They will fight for most of the pots they are in and are fearless opponents. While NITs are risk-adverse, LAGs do not fear risky situations; rather, they embrace them. When a LAG is in a hand, they put pressure on their opponents and aren’t afraid to bluff and re-raise with the worst hand in the right spots. It’s important to note that LAGs don’t have uncontrolled aggression at the table, like their bad aggressive counterparts. Actually, the opposite is true. LAGs use controlled aggression to put their opponents into tough spots, knowing how and when to bluff as well as how to effectively value-bet to get maximum value.

There are three basic types of bad poker players:

• Loose Passive
• (Loose Passive) Calling Stations
• Bad Aggressive (Maniacs)

• Loose Passive

• A loose passive opponent type is the stereotypical bad player. As the name indicates, they are quite loose and passive as they play. A loose passive opponent loves to limp in pre-flop to try to see flops for as cheap as possible. However, when facing pre-flop aggression, a loose passive opponent will usually fold. This type of opponent plays in a fit-or-fold manner post-flop, meaning they will fold if they miss out and will often never bluff. A loose passive opponent will only bet or raise pre-flop and post-flop with a strong hand or very strong draw. When you play against a loose passive opponent, you will see him limping in pre-flop a majority of the time. A passive opponent will only raise pre-flop with the top of his starting hand range. This type of opponent is very common at the online micro stakes and live low stakes.

• A calling station is a type of loose passive opponent. They share many of the same characteristics, except for one crucial difference: calling stations hate to fold. Calling stations love to limp and see flops, but tend to not fold to aggression, making them almost impossible to bluff. They will call pre-flop, even to raises and re-raises with a wide range of hands. Post-flop, they will float continuation bets with draws and ace-high hands, but just like their loose passive counterpart, they will usually only become aggressive and bet or raise with a very strong hand.

• The bad aggressive opponent, commonly referred to as the maniac is the bad player version of the LAG. While a LAG can control their aggression, bad aggressive maniacs have uncontrolled aggression. They love to gamble by betting and raising relentlessly without any sound strategy in mind. Most bad aggressive maniacs will have a huge stack, be down multiple buy-ins, or bust out of the table very quickly. You will often see huge swings in their chip stacks in a relatively short period of time. Because they have uncontrolled aggression, you can never tell exactly what they have, and they could either be bluffing or value-betting. Moreover, they tend to put people on tilt when they make silly moves and suck out, taking down a huge pot. The great thing about bad aggressive opponents, though, is that they can be easy targets to double up against if you play against them correctly.

Probability of Being Dealt Pocket Aces

• Since there are 4 Aces in a deck (A♣ A♦ A♥ A♠), the probability of being dealt one Ace is 4 in 52. Once we’re dealt one Ace, there are now only 3 Aces left in the deck of 51 remaining cards; therefore, the odds of our second card also being an ace is 3 in 51. We combine these two probabilities together,to get a 0.452% chance of being dealt pocket Aces
  • (4/52) x (3/51) = 0.452% Probability.
• This probability holds true for any poker pair if you are asking the probability of being dealt a “specific” pocket pair before the hand is dealt by the dealer.

Probability of Being Dealt Any Two Suited Cards

• Now let’s determine the probability of being dealt any two suited cards. In this scenario, the first card doesn’t matter because whatever we’re dealt first, we need the second card to match that suit. Therefore, since we’re always going to be dealt a random first card, all we need to know is the probability of the second card being the same suit as the first. We know there are 13 cards per suit in a deck. Since we have already been dealt one card of that suit, there are 12 remaining in the deck. Put simply, since we started with 13 cards and removed 1, there are 12 left of that suit in the 51 available cards, so there is a 12/51 probability that we’ll be dealt any two suited cards
  • (12/51) = 23.53% Probability

Other probabilities

• We will make our straight 17.02% of the time on the turn and miss it the remaining 82.98% of the time. Lots of people tend to erroneously overestimate the probability of making straight and flush draws
  • The probability of flopping a set or better is 11.76% or 1 in 8.5 times
  • We can also express this is 7.5-to-1 odds, usually written as 7.5:1 odds
  • 2:1 Pot Odds → Reward:Risk Ratio i.e You risk 1 to win 2
• Let’s Convert 2:1 Drawing Odds
  • Given 2:1 → m:n, where m = 2 & n = 1
    • Percentage = n / (m + n)
    • Percentage = 1 / (2+1) = 1/3
  • 1/3 then reduces to 33.3% odds
  • So 2:1 drawing odds is equal to 33.3% drawing odds

<Odds ratios to odd %es image>

Equity

• What does equity mean? Equity is our share of the pot if a hand is played to showdown. It tells us how much we expect to win in the long-run based upon how often we should win.

• Let’s use a simple coin-flip example to demonstrate the concept of equity. When you flip a coin and choose either heads or tails, you expect either heads or tails to hit 50% of the time over the long-run. In other words, if you pick tails and wager on it, you expect to win 50% of the time. Therefore, you have a 50% equity, or chance of winning, in a coin-flipping wager.

• So if you wager $1 on a coin flip, you expect to win $0.50 in the long run. Why? Your equity is 50% of the pot:
  • Your Coin Flip Equity: $1 Wager x 0.50
  • Probability of Coin Landing on Tails = $0.50
• Therefore, your equity can be expressed as a percentage or a dollar amount:
  • Percentage Equity: 50%
  • Dollar Amount Equity: $0.50

The Equity Caveat: Variance

• There is a caveat to equity; it is a long-term expectation.

• What in the world does that mean?

• It means that mathematical variance can cause significant, unexpected results in the short-term, where your actual winnings and losses don’t match your expected equity outcome. Variance occurs when there are deviations from expected results. For example, you could flip a coin 4 times in a row and have it land on tails 100% of the time. This would be considered short-term variance, since we expect to hit tails only 50% of the time.

• We’ve all run into sessions where we were a huge favorite with pocket Aces or Kings pre-flop only to get sucked out on and lose with them several times in a row. This is a classic example of variance in poker. If you take poker seriously and play tens of thousands to hundreds of thousands of hands per a year, variance will play a huge role in unexpected upswings and downswings. What you’ll notice is that variance tends to be magnified over smaller sample sizes and minimized as you play more and more hands.

• Pre-flop all-in situations are a common occurrence in poker, so we’ll use a fairly common scenario of QQ versus AK all-in pre-flop. In this situation, QQ is a 55% favorite to win, meaning QQ has 55% equity whereas AK has the remaining 45% equity. Let’s assume the all-in pot size is $200 and determine QQ and AK’s equity in dollar amounts:
  • Dollar Amount Equity = % Equity x Pot Size
  • QQ Equity = 0.55 x $200 = $110 Equity
  • AK Equity = 0.45 x $200 = $90 Equity
• In the long run, QQ’s 55% equity share of the pot will yield $110 in this all-in situation, whereas AK’s 45% equity will yield only $90. While this is commonly called a “coin flip” scenario, QQ actually wins $10 and AK loses $10 in the long run each time this situation occurs.

- One very important concept to understand when it comes to equity is that equity changes throughout a hand. What does that mean? Our pre-flop equity isn’t the same as our post-flop equity. Pre-flop, we don’t know what cards are going to hit the flop, turn and river. But as they do, our equity in the hand changes accordingly. - We have T♦ T♠ on the BTN. We open raise to 2.5bb, a bad aggressive 20bb short-stacker in the SB 3-bet jams all-in and the BB folds. We have seen the SB open jam with all pocket pairs, stronger suited connectors and all sorts of Broadway cards, so we estimate his all-in jamming range looks like this: 22+, ATs+, KTs+, QTs+, JTs, T9s, 98s, 87s, 76s, ATo+, KTo+, QTo+, JTo. What is our equity against this range? Using Equilab, we determine that T ♦ T ♠ is a 60.54% equity favorite against the bad aggressive SB’s all-in jamming range, which can be seen below. This means that we expect to win 60.54% of the time in the long-run against SB’s range of hands in this example. If we call, our rightful share of the 41.5bb pot is 60.54% of the entire pot Our 60.54% Equity Share : 41.5bb x 0.6045 = 25.12bb. - Should We Call? The bigger question is, “Should we call?” The answer is yes, based upon the pot odds we’re being offered combined with our equity chance of winning the hand; however, we haven’t learned about evaluating pot odds versus equity yet. We will decide whether to call or not based on pot odds. - We called a pre-flop raise in the CO with J♠ T♠. The flop comes 8♠ 9♥ A♠. We estimate that our opponent has either AK, AQ or AJ. What is our equity in this situation? According to Equilab, with both an open-ended straight draw and flush draw, our hand currently is a 53.82% equity favorite to win the hand by the river against AK, AQ or AJ. - A LAG open-raises UTG for $4 in a $1-$2 No-Limit game at our local card room. It folds around to us in the BB and we call with 9♥ 9♣. The flop comes 2♠ 7♠ 8♦. We check, villain bets a 1/2 pot-sized bet and we call. The turn is the 3♣. Again we check, villain bets a 1/2 pot-sized bet and we call. The river is the T♥, making the final board 2♠ 7♠ 8♦ 3♣ T♥. We check, villain bets a $40 pot-sized bet and the action is on us. What is villain’s perceived range and our equity against that range? This is an interesting spot, especially for beginner poker players. The first thing we need to take into consideration is villain’s playing style. Villain is an aggressive LAG, so we know he’ll have a wider UTG open-raising range than most other decent players. He’ll also put immense pressure on us post-flop, especially since we’re out of position and without the initiative. A good assumption would be around a 16% opening range: 22+, ATs+, KTs+, QTs+, JTs, T9s, 98s, 87s, ATo+, KJo+, QJo. This range gives villain a plethora of made hands and busted draws to triple barrel into us on the river. The missed draws we’d expect villain to triple barrel into us are missed flush draws, Broadway over cards looking to hit a pair and missed straights. Villain also has a lot of made hands, such as over pairs, top pair, two pair sets and straights. Lastly, villain could be bluffing with a small pair, such as 22-66. Usually we would narrow down our opponent’s range on each street of action, but we shouldn’t be surprised to see such a good, aggressive opponent bet three streets against us with his entire starting range to put pressure on us. A good LAG knows it will be hard for us to call three streets of bets with a single pair when we have shown weakness by taking a check/call line for three streets post-flop, so the question remains, how are we doing with our 9♥ 9♣ against villain’s entire range? Plugging this information into Equilab, we see that we are actually a 61.66% equity favorite against villain’s entire range in this spot. Since our opponent has so many bluffs in his range – as well as missed draws – our measly pair of nines is doing well against his entire range of hands on the river in this spot. If we call, we expect to win 61.66% of the final $80 pot over the long run, which equates into $49.33 - All in all, equity is our rightful share of the pot and we can use equity tools such as Equilab to perform complex equity calculations off the table **Pot Odds** - Pot odds are the immediate odds we’re being offered when we call a bet in poker. The important aspect of this definition is immediate, because with pot odds it’s all about how much we stand to win immediately in relation to what we are risking by calling a bet. Remember, this relates directly to the reward-to-risk ratios - For example, if our opponent bets “x” amount in a hand on the river, we are given a certain amount of pot odds to make the call on the river in order to potentially win the pot. With our reward:risk ratio, we risk the amount we have to call in order to win the reward of the amount of money in the pot - Example: There’s $10 in the pot going to the river. Villain bets all-in for a total of $10 and the action is on us. We can either call or fold in this spot. If we call, we are risking $10 to win the $20 already in the pot, so our reward:risk ratio is $20:$10 or 2:1. Mathematically, we can convert this into a percentage and say we are calling $10 out of a total of $30 ($20 pot + our $10 call), which would mean we have to put 1/3 or 33% more into the pot to get $30 back. So, in this spot, we are getting the following pot odds: $20:$10 →2:1 → 1/3 → 33% Pot Odds - Should we call? If we expect to win at least 33% of the time, we should call. Determining if we can profitably call or not is a function of comparing our pot and implied odds to our equity in the hand - Pot odds = [pot size]:[amount to call] where [pot size] includes any and all bets on the current street (pre-flop, flop, turn, or river) not including our call - When determining our pot odds, we are concerned with the total [pot size], not including our call, and the [amount to call] The following is the second way to determine pot odds as a percentage: Pot Odds % = call size / (pot size before call + call size) - When determining our pot odds percentage, we include our call amount into the total pot size, because we’re calculating how much we must call into the total amount that will be returned to us if we win **Ratio to Percentage Method** - Pot Odds = [pot size]:[amount to call] = [12bb + 8bb + 20bb]:[20bb] = 40bb :20bb = 2:1 Pot Odds = 2:1 Convert Ratio to Percentage = 2:1 → 1 / (2+1) → 1/3 → 33% Pot Odds - We use pot odds to ensure we only call bets when we are getting good pot odds or good implied odds based upon our equity in the hand. In gambling and in poker, there are good odds and bad odds. Good odds favor us in the long-run, whereas bad odds favor our opponent(s) **Implied Odds** - Implied odds state that we can call a bet now, even if we are not getting good direct pot odds, if we expect to make up for it on later streets (turn or river) of betting if we hit our draw. What this means is that we can call a bet now, getting bad direct pot odds with a drawing hand, if we expect our opponent to pay us off nicely when we hit our draw. So, implied odds reflect how much we expect to win on later streets when we hit our drawing hand - Good Implied Odds: If we expect to win a lot more money from our opponent on later streets of betting after we make our draw, we have good implied odds - Bad Implied Odds: If we expect to win little or no more money from our opponent on later streets of betting after we make our draw, we have bad implied odds - Concerned with how much we will win now, as well as how much we stand to win on later streets of betting action when we make a drawing hand. We don't need a good direct pot odds price to call a bet based upon our equity if we expect to get paid off nicely on later streets of action after we make the best hand - Implied odds work well: - Against Aggressive Opponents - Against Calling Stations - When You’re in Position - In Multi-Way Pots - When You’re Deep-Stacked - With Hidden Draws - Against aggressive opponents, we can call flop and turn bets with bad direct pot odds when we expect our aggressive opponent to pay us off when we hit our draw. Implied odds are higher with aggressive opponents because they will usually tend to bet on later streets for value, and as a bluff even when we hit our draw. These types of opponents will put immense pressure on us post-flop, doing their best to try to make us fold the best hand. Since they aren’t afraid to bluff, these types of opponents – especially bad aggressive maniacs – typically yield us good implied odds. - Loose passive calling stations are great opponents to get paid off on later streets when we make the best hand. Since they hate folding post-flop and will call river bets with as little as bottom pair, we can look to extract implied odds value when our drawing hands hit on the turn or river. Moreover, when a loose passive calling station takes the lead in a hand by raising pre-flop and continues betting the flop, we can easily put them on a strong range of made hands that they’ll commit to on the turn or the river. While we don’t want to try and bluff raise or bluff bet them out of a hand, we can profitably look to take a check/call line to try to hit the best hand on the turn or river - It’s always easier to extract value when we’re in position, because we’re last to act in each round of betting. Being last to act ensures us the ability to wager a bet when we hit our draws. Accordingly, this allows us to get paid off more often than when we’re out of position. For example, if we’re out of position, hit our draw, and check to our opponent, our opponent may check behind for pot control. However, if we’re in position, we can fire out a value bet and hope to get called by our opponent. Because of this, it makes much more sense to play implied odds, drawing hands more often in position than out of position because of this opportunity to act last post-flop - Multi-way pots with multiple opponents in a hand often yield good implied odds. Hands that can hit hidden monster draws such as sets or hidden straight draws in multi-way pots are a great way to extract enormous amounts of implied odds winnings. The reason for this is fairly straight forward. The more people there are in the hand, the more likely our opponent(s) will make a strong made hand or strong draw. This increases the likelihood that we’ll get paid off with our very strong holdings - When effective stack sizes in a hand are deep-stacked, we’re also more likely to get paid off in implied odds situations. The main reason is that when SPRs are large, we have a lot of room for post-flop maneuverability with large chip stacks compared to the size of the pot. Situations such as 3-bet pots are ideal to play starting hands that can make monsters such as sets, straights and flushes against an opponent who’s showing tremendous pre-flop strength for this very reason - What is a hidden draw? It’s typically a set or straight using one-gapper connectors. These are draws that are very difficult to detect; therefore, we can expect to yield a lot of value when we hit them. Here are some examples: - We have 6♣ 8♣ and the Flop is 5♣ 7♦ K♥ - We have 9♥ 9♣ and he Flop is K♣ 2♠ 8♦ - If we hit the straight on the turn with a 9 with 6♣ 8♣, it will be very difficult for our opponent to see the straight. If our opponent has top pair, he will definitely continue to bet for value on the turn. The same goes for the second example. If we turn a set of 9’s, our opponent will never see it coming and will continue to value bet top pair. These are classic examples of hidden draws that have a lot of implied odds value because they are so difficult to spot. With hands like these, we can expect to extract maximum value when they hit - Most calling stations are unaware of anything other than their own holding; therefore, they’ll be unlikely to see that we hit a flush on the turn or the river and will only be playing their own hand - Five different situations in which implied odds don’t work too often: - Against Tight Opponents - Against Short-Stackers - End of Action Spots - When You Are Out of Position - With Obvious Draws - Knowing how much implied odds we can extract from our opponents when we make the best hand will allow us to make the most optimal decision possible when we call an initial bet getting a bad pot odds price. Lots of inexperienced poker players will simply look at their opponent’s stack size and declare to themselves that they’ll be able to stack their opponent if they hit their draw. This is an erroneous conclusion, as we never fully know how much money we’ll be able to extract from our opponents when we hit our drawing hands. Furthermore, a lot of factors feed into how much we’ll be able to extract to determine if we’re getting good or bad implied odds - Below are some common draws you’ll often see. You should memorize their corresponding outs: - Flush Draw: 9 Outs - Open-Ended Straight Draw (OESD): 8 Outs - Over Cards 2-Pair Draw: 6 Outs - Gut-Shot Straight Draw: 4 Outs - Sometimes certain outs that complete our draw also complete our opponents’ draw to a better made hand. Furthermore, sometimes our opponents will have some of our potential outs in their possible range of hands. When this occurs, we should discount those outs, meaning we should not include them in our outs calculation. Whenever we think our outs to a draw are already taken by one of our opponents, or if we don’t expect to win the hand if we hit our out, we should discount them. - We have K♠ Q♠ and flop an open-ended straight draw on a J♥ T♥ 2♠ board texture. In this situation, we should, at minimum, discount the A♥ and 9♥, because while these two cards make our straight, they also make a potential flush for our opponent. Furthermore, if we think our opponent has a lot of Ax hands in his range, including non-heart Aces such as AQ, AJ and AT, we should discount additional Aces from our potential outs to make our straight - When evaluating board textures, we need to be able to identify when an out is a dirty out, meaning it completes a better draw for one of our opponents. When we identify a dirty out, we should err on the side of caution and discount it. Many poker players fail to discount dirty outs, which leads them to overestimating their drawing hand equity and making bad, unprofitable calls - We have Q♦ J♠ and the flop is 4♥ 7♠ 4♦. Our Draws On this flop board texture, our only draw is a pair of Queens or Jacks on the turn or the river with our over cards. We don’t have any other draws. Our Outs: We don’t have to worry about discounting any of our outs with our over card pair draw. We have a total of 6 combined outs: 3 Queens 3 Jacks We have A♠ T♣ and the flop is Q♠ J♣ 6♣. Our Draws On this flop board texture, we have a gut shot straight draw to the nut straight and an over card draw to a pair of Aces. Our Outs Since there is a flush draw on this flop board texture, we must first eliminate all clubs from our potential outs, because if any club hits, it will complete a flush that will beat our straight or pair of Aces. So, how many outs do we have to make our straight or pair of aces? Straight Draw : K♦, K♥, K♠ Pair of Aces : A♦, A♥ Discounted (Eliminated Outs) : K♣, A♣ Total Outs : 5 Outs We have 7♣ 7♥ and the flop is T♦ 5♠ 2♠. Our Draws If we assume our opponent has a pair of tens, we are only drawing to a set of sevens. Our Outs Although there are two sevens left in the deck, we don’t want to see a spade because our opponent could also have a flush draw. We should eliminate the 7♠ from our outs, leaving us with one out; the 7♦. We have A♠ 5♠ and the flop is 7♠ 7♣ T♠. Our Draws We have a draw to the nut flush and if we assume our opponent doesn’t have trip sevens or a full house, we also have a draw to a pair of Aces. However, if our opponent does have trip sevens, then our only draw is the flush. If our opponent flopped the full house, we are drawing dead, meaning we cannot win the hand. Our Outs We don’t have to discount any outs, since none of our outs improve our opponent’s draw unless our opponent has a stronger Ace. Let’s assume our opponent rarely has trips or the full house. With that in mind, we have the following outs: Flush Draw : 9 Outs Pair of Aces: A♦, A♥, A♣ Total Outs : 12 Outs We have K♦ J♣ and the flop is J♦ 5♦ A♦. Our Draws We have a plethora of draws in this hand if we think our opponent has a pair of Aces. So what are our draws? We have the nut flush draw, trip jacks draw, and a two-pair draw. Our Outs Let’s discuss our outs for each draw: Flush Draw : 9 Outs Trip Jacks : J♥, J♠ Two-Pair Draw : K♣, K♥, K♠ Do we need to discount any outs? We don’t need to discount outs for the nut flush draw. However, if it’s likely our opponent flopped a smaller flush, a set, or AJ for top two pair, we need to discount our trip jacks and two-pair draws. We shouldn’t assume our opponent has such a strong hand all of the time, so we don’t need to discount our outs for trips and two-pair completely. It would be safe to discount them by approximately 50% and assume we only have 2 or 3 outs instead of 5 for those two draws. So, with discounting our outs, we can conservatively assume we have 11 or 12 outs rather than the initial 14 outs we calculated The Rule of 2 and 4 is very simple and involves two simple steps: Step 1: Count our outs for our draws Step 2: Multiply our draws by 2 or 4 based upon the criteria below: Multiply Outs x 2 on the flop if we’re not being put all-in or calling an all-in. Multiply Outs x 4 on the flop if we’re being put all-in or are calling an all-in. Multiply Outs x 2 on the turn no matter the circumstance <2 and 4 rule for popular draws> <2 and 4 rule for all outs> We have Q♦ J♠ and the flop is 4♥ 7♠ 4♦. Draw(s): Pair of Queens or Jacks Outs: 6 Outs Rule of 2 & 4 Flop All-In Equity: 6 x 4 = 24% Equity Flop Not All-In Equity: 6 x 2 = 12% Equity Turn Equity: 6 x 2 = 12% Equity A NIT open-raises to 3bb from UTG. A loose passive opponent in MP calls, CO folds and the action is on us with T♠ 9♠ on the BTN. We call, both of the blinds fold and the pot is 10.5bb going to the flop. The flop is A♦ T♦ 2♣, UTG fires out a 2/3 pot-sized continuation bet, MP calls and the action is on us once more. What do we estimate our outs and equity are in this situation, taking into consideration our opponents’ ranges? Given that the initial raiser is a NIT open-raising from UTG, we would expect this opponent to have a very strong UTG opening range. Furthermore, we wouldn’t expect a NIT to fire out a continuation bet into two opponents without a strong made hand or a draw to the nuts. Therefore, we should weigh UTG’s range as being heavily composed of strong Aces, such as AK, AQ and AJ, sets of Aces and Tens, as well as the nut flush or nut straight draw. MP on the other hand, being a loose passive opponent, will not have nearly as strong a range, either pre-flop or on the flop, and therefore we expect MP to call with a very wide range of hands pre-flop. Accordingly, we would expect MP to have a wide variety of Aces, such as AQ-A2, straight draws and flush draws on the flop. Given this information, our outs to improve to the best hand are very minimal in this situation. If we assume that our opponents only have top pair, then non-diamond tens and nines will improve our hand to trips or two pair: T♣, T♥, 9♣, 9♥, giving us a total of 4 outs. If we then assume our opponents will have 2-pair or better a small percentage of the time (approximately 25% or less), we should discount one of our outs, effectively giving us 3 outs to potentially win the hand. Our Draws & Outs Draw(s): Trips and Two-Pair. Outs: 3 Outs Rule of 2 & 4 Flop Not All-In Equity: 3 x 2 = 6% Equity A TAG open-raises to 3bb from the CO position, BTN folds, SB calls, and we call from the BB with 8♦ 8♥, making the pot 9bb going into the flop. The flop is 9♣ 4♠ 2♥, SB checks, we check, and CO continuation bets 5bb. SB folds, we call and the pot is now 19bb going to the turn. The turn card is J♦, we check again, CO bets 12bb and the action is on us. What do we estimate our outs and equity are in this situation, taking into consideration our opponents’ ranges? We expect a good tight aggressive opponent to open a fairly wide range of hands pre-flop from late position. So when villain fires out a continuation bet on the flop on a 9-high board, we shouldn’t give him too much respect. Many times, he’s bluffing with over cards, trying to get us to fold better hands. With a pair of 8’s, we’re fairly certain we have the best hand a decent percentage of the time on this flop board texture, so we easily make the call. The turn is where things get a bit dicey. When we call villain’s flop continuation bet, he must assume we have a pair or Broadway over cards. Given this information, we shouldn’t expect a TAG to double-barrel bluff the turn too often. When the Jack hits the turn and villain fires out a second bet, we no longer feel so great about the prospects that we have the best hand. It’s possible villain has a pair and is betting for value, specifically a pair of 9’s, T’s, J’s or better. It’s also possible that villain has a hand such as KT or QT with the straight draw. Given this information, we expect to be behind a lot, with the worst hand. If this is the case, the only way for us to win this hand is to improve to a set of 8’s on the river. Our Draws & Outs Draw(s): Sets Outs: 2 Outs Rule of 2 & 4 Turn Equity: 2 x 2 = 4% Equity A TAG open-raises to 3bb from the BTN, SB folds, and we 3-bet to 11bb with A♣ A♠ from the BB. BTN calls our 3-bet, and the pot is 22.5bb going to the flop. The flop is A♥ J♥ T♥, we bet 18bb, and BTN raises to 45bb. What do we estimate our outs and equity are in this situation, taking into consideration our opponents’ ranges? This is a very interesting spot. Given the way the hand is playing out, we are either way ahead with the best hand or way behind with the worst hand. Pre-flop, when we 3-bet a good tight aggressive opponent’s BTN steal attempt from the BB, BTN will assume we are potentially defending our blinds with a polarized range composed of strong value and bluff hands. Given this, we should expect BTN to call pre-flop with a fairly wide range, excluding hands he would 4-bet: JJ-33, AQs-ATs, KTs+, QTs+, JTs, T9s, 98s, 87s, 76s, 65s, ATo+, KTo+, QTo+, JTo. Against this range, we’re an equity favorite. However, against a flop raise range, we aren’t doing so well. Hands we’d expect BTN to raise the flop with are the nut flush draw, nut flush + nut straight combo draws, smaller made flushes, made straights and sets. Furthermore, we would also expect BTN to call with all other hands, including 2-pair combos. Given this information, and for the sake of this exercise, let’s assume we’re behind to the following range (even though realistically BTN will have a combination of hands that are losing to us as well as those he’ll raise, such as sets and the nut flush draw): Medium to Small Made Flushes : 9♥ 8♥, 8♥ 7♥, 7♥ 6♥, 6♥ 5♥ Made Straights : KQo If we assume that we’re behind made flushes and straights, we’re only drawing to a full house or better to win the hand. Assuming BTN doesn’t have sets in his range, then only an Ace, Jack or Ten will improve our hand: A♦, J♦, J♠, J♣, T♦, T♠, T♣. This gives us a total of 7 outs. Our Draws & Outs Draw(s): Full House or Quads Outs: 7 Outs Rule of 2 & 4 Flop All-In Equity: 7 x 4 = 28% Equity Flop Not All-In Equity: 7 x 2 = 14% Equity EV is how much we expect to win or lose on average, over the long run based upon a specific scenario in poker. Every single situation and scenario in poker has an expected value associated with it, with certain situations being profitable (+EV) while others being unprofitable (–EV). Some plays will win us money, while others will lose money It’s important to note that EV is concerned with how well a certain play will do over the long-run, and not the outcome of a single hand. For example, if we were evaluating the EV of calling a pre-flop all-in with pocket Aces against an opponent that has pocket Queens, we know that this is a long-term profitable play, regardless of whether we lose that hand or not. We may get unlucky and lose the hand, but we know that, over the long-run, it’s a +EV profitable play Fundamental poker math is at the core of maximizing our winnings and minimizing our losses. In short, winning poker is +EV poker, whereas losing poker is –EV poker Three simple steps that you can easily implement into your game to determine whether or not you can profitably call a bet with a drawing hand:   First, we determine our pot odds and implied odds. Second, we determine our equity in the hand. Third, we compare our pot and implied odds with our equity to determine if calling is +EV or –EV Ratio Method : We can call if the pot odds are greater than our equity odds (odds of completing our draw): Call : Equity Odds Ratio Less Than Pot Odds Ratio Fold : Equity Odds Ratio Greater Than Pot Odds Ratio   Call Example : 2:1 Equity and 3:1 Pot Odds Fold Example : 5:1 Equity and 2:1 Pot Odds Percentage Method: We can call if the % chance of making our hand is greater than the % of the pot we have to call: Call : Equity % Greater Than Pot Odds % Fold : Equity % Less Than Pot Odds %   Call Example : 33.3% Equity and 25% Pot Odds Fold Example : 16.7% Equity and 33.3% Pot Odds Conversely, when the amount we have to commit to the pot is greater than our chance of making our hand, we have to fold if we also have poor implied odds In our previous example, we have to commit an additional 33.3% to the pot to see the next card, but only expect to make our hand 16.7% of the time. The maximum pot odds we can call in this situation with poor implied odds is 16.7%. Since 33.3% exceeds the maximum pot odds price we can call by 16.6%, this is a mandatory fold without good implied odds We have A♣ T♣ and the flop is 5♣ K♣ 8♥. Villain bets $10 into a $50 pot. Can we call based on pot odds alone? What are the pot odds? For our first exercise hand, I’ll show the pot odds ratio method, pot odds ratio-to-percentage conversion, and pot odds % method as a refresher. Pot Odds Ratio Method = [pot size]:[amount to call] Pot Size = $50 Pot + $10 Bet = $60 Pot Odds Ratio = $60:$10 = 6:1 Pot Odds Pot Odds Ratio to % Conversion = 1/(6+1) = 1/7 = 14% Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = $50 Pot + $10 Bet + $10 Call from Us = $70 Pot Size % = $10 / $70 = 14% Pot Odds How many outs do we have? We have 9 clubs for our Ace-high flush draw. Additionally, if we put our opponent on a pair of Kings, we have an additional 3 outs to make a pair of Aces. This gives up a total of 12 outs. Draw(s): Flush Draw and Pair of Aces Outs: 12 Outs What is our estimated equity? Using the rule of 2 and 4, we multiply our outs by 2, since we are not calling an all-in on the flop: 12 outs x 2 = approximately 24% Equity (actual equity is 25.5%) Should we call? Yes! We have approximately 24% equity in the hand and only have to put 14% more into the pot. Since our equity % chance of hitting our draw is greater than the pot odds % we have to call - this is a +EV call Let’s use the same exact hand, but change villain’s bet size and pot odds. We have A♣ T♣ and the flop is 5♣ K♣ 8♥. Villain bets $40 into a $50 pot. What are the pot odds? Pot Odds Ratio Method = [pot size]:[amount to call] Pot Size = $50 Pot + $40 Bet = $90 Pot Odds Ratio = $90:$40 = 2.25:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = $50 Pot + $40 Bet + $40 Call from Us = $130 Pot Size % = $40 / $130 = 31% Pot Odds How many outs do we have? We have 9 clubs for the flush draw & 3 Aces for an over pair draw, giving us a total of 12 Outs. Draw(s): Flush Draw and Pair of Aces Outs: 12 Outs What is our estimated equity? Using the rule of 2 and 4, we multiply our outs by 2 and 4: 12 outs x 2 = approximately 24% Equity (actual equity is 25.5%) 12 outs x 4 = approximately 48% Equity All-In (actual equity is 45%) Should we call? It depends on our implied odds, or if we’re willing to raise all-in on the flop. Based upon direct pot odds, we should fold, since we have to put 31% more money into the pot and only expect to make our draw 24% of the time. Based upon direct pot odds alone, this would be a –EV call. However, if we consider raising all-in on the flop, we ensure that we get to see both the turn and river card without putting additional money into the pot. Raising all-in as a semi-bluff improves our equity to 45%; while simultaneously providing additional benefits. If villain only has a pair of Kings, we can make him fold better hands by forcing him into a tough all-in decision. By semi-bluff raising, we can now win the hand by either making our opponent fold, or making the best hand on the river. Let’s now consider implied odds. If we think that villain will pay us off if we hit our Ace or flush, then we can call, because we have good implied odds. However, if we think he’ll shut down and not put any more money in the pot when we hit the winning hand, we should just fold, or go all-in on the flop, since we have bad implied odds. Calling or raising all-in is villain-dependent in this situation, and we should only call if we think we have good implied odds We have 7♦ 9♦ and the turn is 6♦ 8♠ A♦ 2♣. Villain goes all-in for $3.50, making the pot now $7.50. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $7.50 Pot Odds Ratio = 7.50:3.5 = 2.14:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = $7.50 Pot Size + $3.50 Call from Us = $11.00 Pot Size % = $3.50 / $11.00 = 32% Pot Odds How many outs do we have? Assuming our opponent is going all-in on the turn with a pair of Aces, two pair or a set, we’re drawing to either a flush or straight to win the hand on the river. Flush Draw : 9 Outs Open-Ended Straight Draw : 6 Outs (2 outs for this draw are accounted for in the flush draw, 5♦ and T♦) Outs: 15 Outs What is our estimated equity? Rule of 2 & 4: Turn Equity: 15 outs x 2 = 30% Equity (actual equity is 31.9%) Should we call? Yes, we should call. We are getting 32% pot odds and expect to hit our draw just under 32% of the time, so we’re getting the correct pot odds to call here with both a flush and open-ended straight draw. Remember that since this is an all-in situation, there are no implied odds We have K♠ Q♠ and the flop is A♠ 9♠ 3♥. Villain bets $2.00 into a $2.50 pot, we raise to $6.50 and villain re-raises all-in for a total of $10.00. The pot is now $19.00 and we have to call an additional $3.50 with our draw if we want to continue. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $19.00 Pot Odds Ratio = 19:3.50 = 5.42:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = $19.00 Pot Size + $3.50 Call from Us = $22.50 Pot Size % = $3.50 / $22.50 = 15.6% Pot Odds How many outs do we have? At minimum, villain is going all-in in this situation with at least a pair of Aces, so we’re drawing to get only the nut flush draw to win this hand. Draw(s) : Flush Draw Outs: 9 Outs What is our estimated equity? Rule of 2 & 4: Flop All-in Equity: 9 x 4 = 36% Equity (actual equity is 35%) Should we call? Yes, this is an easy call. We’re getting excellent pot odds to call. We only have to put 15.6% more into the pot and expect to win 36% of the time versus villain’s all-in jam on the flop. Folding here would be a huge mistake We have K♠ T♥ and the flop is A♥ Q♣ 4♣. Villain bets $5.00 into a $4.00 pot. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $4.00 Pot + $5.00 Bet = $9.00 Pot Odds Ratio = 9:5 = 1.8:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = $9.00 Pot Size + $5.00 Call from Us = $14.00 Pot Size % = $5.00 / $14.00 = 36% Pot Odds How many outs do we have? When our opponent over-bets a flop board texture like this, it’s indicative of a scared recreational player over-betting a pair of Aces – or better, trying not to get sucked out by the flush or straight on the turn. With a bet like this, their goal is to usually bet us out of the hand to protect their made hand and scoop the current pot. They aren’t looking to extract value. Instead, they’re purely looking to protect their hand. Knowing this, we really shouldn’t be afraid of the flush draw in a heads up pot in this particular situation, so we don’t really need to discount the J♣. This gives us a total of 4 outs to the straight. Draw(s) : Gut shot Straight Draw Outs: 4 Outs What is our estimated equity? Rule of 2 & 4: Flop Not All-in Equity: 4 x 2 = 8% Equity (actual equity is 8.5%) Should we call? It depends. We only have 4 outs to hit the nut straight. This gives us 8% equity, but we’re being forced to put 36% more into the pot. This is a –EV pot odds call. So, based purely upon pot odds, this is a clear fold. But, if we suspect we have excellent implied odds, we might consider making the call. Let’s evaluate our implied odds. Clearly our opponent is afraid of the obvious flush draw hitting the turn. Based on our assumptions of villain, we can assume that if the J♣ comes on the turn, we won’t get paid off, because it’s an obvious scare card for our opponent. However, if a non-club Jack comes on the turn, it’s not nearly as much of a scare card for villain. If villain has a two-pair, a pair of Aces with a good kicker, or a set, he’s likely to continue with his aggression on the turn. The benefit of our inside straight draw is that it’s a hidden draw. With that being said, if effective stack sizes are deep enough and a non-club Jack hits the turn, we have good implied odds. So, calling based upon pot odds alone is a –EV play. But, if we assume our implied odds are excellent, then we can consider calling in this spot A bad aggressive opponent, who has been playing wildly throughout the session, bluffing relentlessly, open-raises to 5bb in MP. With 100bb effective stack sizes and 7♥ 8♥ in the CO, we make the call. Everyone else folds and the pot is 11.5bb going to the flop. The flop is 6♣ T♦ A♥ and UTG continuation-bets a pot-sized 11.5bb bet. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = 11.5bb Pot + 11.5bb Bet = 23bb Pot Odds Ratio = 23bb:11.5bb = 2:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = 23bb Pot Size + 11.5bb Call from Us = 34.5bb Pot Size % = 11.5bb / 34.5bb = 33% Pot Odds How many outs do we have? We are drawing to a gut-shot straight draw, giving us 4 outs. Draw(s) : Gut-Shot Straight Draw Outs: 4 Outs What is our estimated equity? Rule of 2 & 4: Flop Not All-in Equity: 4 outs x 2 = 8% Equity (actual equity is 8.5%) Should we call? With only a gut-shot straight draw and 8% equity, we don’t have nearly enough equity to call MP’s continuation bet purely off 33% pot odds alone. Making a call off pot odds alone would be an unprofitable play. However, our opponent’s playing style and tendencies provide the possibility of getting paid nicely if we hit our draw. With 100bb effective starting stacks, our opponent has 83.5bb left behind in his stack after his pot-sized flop continuation bet. Moreover, if we call MP’s flop bet, there will be 34.5bb in the pot going into the turn. Assuming MP either has an Ace or is purely bluffing, a relatively blank card such as a 9 will not slow him down from firing a second barrel. Since MP open-raised to 5bb pre-flop, a fairly large pre-flop raise, we can also assign him a fairly strong range that includes a lot of Ax hands such as AK, AQ, and AJ. With that being said, this situation can yield very nice implied odds profits on the turn and river against a bad aggressive opponent. If he has an Ace, we can expect him to fire out a large bet on the turn. If he’s bluffing, we can also expect him to bluff a high percentage of the time to put pressure on us to fold. With 83.5bb left in his stack, this is a good implied odds situation to try and hit our straight on the turn and stack our opponent We open-raise to 3bb in MP with K♣ Q♣ with a 135bb stack, it folds around to a 69bb stack calling station in the SB that calls, and we go to the flop heads up. The pot is 7bb going to the flop. The flop is A♣ 7♣ 2♥, SB bets 8bb, and the action is on us. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = 7bb Pot + 8bb Bet = 15bb Pot Odds Ratio = 15bb:8bb = 1.88:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = 15bb Pot Size + 8bb Call from Us = 23bb Pot Size % = 8bb / 23bb = 35% Pot Odds How many outs do we have? We’re drawing to a flush in this situation in order to beat our opponent’s potential pair of Aces. Draw(s) : Flush Draw Outs: 9 Outs What is our estimated equity? Rule of 2 & 4: Flop Not All-in Equity: 9 x 2 = 18% Equity (actual equity is 19.1%) Should we call? This is an interesting spot and how we play it greatly depends upon our opponent. In this situation, our opponent is a loose passive calling station. This type of opponent is very passive, meaning he or she will only bet with a very strong made hand or drawing hand. Since we hold the nut flush draw, SB most likely has top pair, two pair, or a set. We clearly cannot call this bet purely off pot odds alone, given that we only have an 18% equity chance of hitting our flush on the turn with a 35% pot odds bet. With this information, calling for implied odds depends on if we’ll get paid off if we hit our flush. Most bad recreational calling stations are unaware of anything other than their own holdings. They are not likely to see that we hit a flush on the turn or the river, and they’ll only be playing their own hand. When they hit any piece of the board, they’ll often put a lot of money into the pot – that’s why we love calling stations. With a hand that villain is willing to bet into us on the flop where he wasn’t the pre-flop aggressor, we can expect villain to have a very strong made hand. This, combined with a calling station’s inability to fold, skyrockets our implied odds. Knowing this, we have great implied odds in this situation. While effective stacks aren’t deep, we do have a position on our opponent and are likely to get paid off if we hit our flush. What about re-raising all-in on the flop? While this is a decent option in the correct circumstances, re-raising is a semi-bluff in this situation. When we re-raise all-in as a semi-bluff, we need two components. The first is decent fold-equity, i.e., the high likelihood that our opponent will fold to our raise. The second is strong all-in equity. In this situation, our fold equity isn’t great against a calling station, nor do we have amazing all-in equity. We only have approximately a 36% equity chance of making our flush by the river when we’re all-in on the flop. We would prefer to have closer to 50% when we jam all-in on flop situations. Given this information, this isn’t an ideal all-in situation A NIT open-raises to $6 in MP in a $1-$2 No-Limit Hold’em game. We call with K♦ Q♦ in the CO. The BTN folds, SB folds and BB calls. There’s $19 in the pot going to the flop. The flop is A♦ 9♦ 2♠. BB checks, MP fires out a $15 continuation bet, and the action is on us. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $19 Pot + $15 Bet = $34 Pot Odds Ratio = $34:$15 = 2.27:1 Pot Odds Pot Odds % Method = Call Size / (Pot Size + Call Size) Pot Size + Call = $34 Pot Size + $15 Call from You = $49 Pot Size % = $15 / $49 = 31% Pot Odds How many outs do we have? We are drawing to the nut flush. Draw(s) : Nut Flush Draw Outs: 9 Outs What is our estimated equity? Rule of 2 & 4: Flop Not All-in Equity: 9 outs x 2 = 18% Equity (actual equity is 19.1%) Should we call? Since we are being offered 31% pot odds and have an 18% equity chance of making our flush on the turn, this would be a –EV pot odds call; therefore, based upon pot odds, we should fold. Implied odds, in this situation, are poor as well. Playing against a NIT who will shut down at any sign of the flush completely negates our implied odds possibilities on the turn or river. Accordingly, calling this flop bet would be a mistake based upon pot and implied odds combined – making this an easy fold When faced with all in decisions, we need to consider the following: Our Pot Odds Our Opponent(s) All-In Jamming Range Our Equity vs. Our Opponent(s) Ranges Multi-Way Side Pots & Implied Odds In most all-in situations, there are no implied odds. The only time implied odds are considered is when there are three or more opponents in the hand, and at least two of the people aren’t all-in. This frequently happens when a short-stacker goes all-in, and two or more moderate to deep stacks call, but aren’t all-in themselves. When this happens, a side pot occurs, potentially with additional implied odds considerations. However, most all-in situations will occur in heads up battles where a person 3-bet, 4-bet, or 5-bet jams all-in, and the other player is forced to make a decision to call or fold. When this occurs, the player needs to base calling or folding upon the pot odds price they’re being offered. For example, if we are faced with an all-in call or fold decision, we need to ensure our equity chance of winning is equal to or greater than the pot odds price we’re being offered. How do we determine our equity? By estimating our opponent’s all-in jamming range Understanding which hands our opponent is likely to go all-in with helps estimate our equity. We need to make an educated guess and assign our opponent a realistic all-in range. We need to ask ourselves questions such as, “Is our opponent only going all-in with the very top end of his range, with hands such as KK+, AK, or is he jamming all-in with weaker hands such as 99 and AJ?” Narrowing down our opponent’s range is a skill that takes time to master, but we can help our cause by taking good notes and having a good understanding of our opponent’s playing style. Having notes – either written if we play online, or mental if we play live – will give us insight on what specific hands our opponent has gone all-in pre-flop in the past. Additionally, knowing if our opponent is a LAG, TAG, NIT, Loose Passive, etc. will help us to further narrow our opponent’s range. Once we have a good idea, we’ll then be tasked with estimating our equity against our opponent’s range We’ll assume a TAG will call an all-in with AA, KK, QQ and AKs, and assign our opponent’s the following jamming ranges below: Ultra-Tight All-In Jamming Range : AA, KK Tight All-In Jamming Range : AA, KK, AKs Moderate All-In Jamming Range : AA, KK, QQ, AK Loose All-In Jamming Range : AA, KK, QQ, JJ, AK, AQs Ultra-Loose All-In Jamming Range : AA, KK, QQ, JJ, TT, AK, AQ, AJ < Our range vs opponent jamming range, our equity vs opponent equity> A vital concept to understand is that as more people call pre-flop, our equity declines, even with pocket Aces. To show how this works, we'll compare AA versus a single opponent with an ultra-tight all-in jamming range. We'll then compare it against two opponents, one with an ultra-tight all-in jamming range and another with a loose all-in jamming range. Playing a $1-$2 No-Limit Hold’em game, we open-raise to $8 UTG with J♦ J♥, the action folds around to a short-stacker LAG in the SB, who 3-bet jams all-in for $44. The action is on us. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $54 Pot Odds Ratio = $54:$36 = 1.5:1 Pot Odds Pot Odds % Method = Call Size / Pot Size + Call Size Pot Size + Call = $54 Pot Size + $36 Call from You = $90 Pot Size % = $36 / $90 = 40% Pot Odds What is our opponent’s estimated jamming range? Knowing that our opponent is a loose aggressive short-stacker, we can expect him to be 3-bet jamming a wide range of hands, including a hand as weak as 66 and AJ. Pocket Pairs : 66+ Broadway Cards : AJ+, KQ+ What is our estimated equity versus our opponent’s jamming range? Referring to the equity chart previously, we should expect to have good equity versus a very loose jamming range, and in fact, much more than the required 40% equity to call in this situation. How do we know this? Out of our opponent’s entire estimated range, a majority of his jamming range is an equity dog to JJ, which tells us we’re a favorite against most of it: JJ Equity Favorite : JJ-66,AJs+,KQs,AJo+,KQo JJ Equity Dog : AA, KK, QQ We’ll utilize Equilab to provide a definitive estimate: Our Equity : 58% Opponent’s Equity : 42% Should we call? Yes. We estimate ourselves to be an equity favorite in the hand. Given that we only need 40% equity to call getting a 40% pot odds price, this is an easy call, where we know we are a 58% equity favorite in this situation. Playing a $1-$2 No-Limit Hold’em game, we open-raise to $6 UTG with 7♦ 7♥, the action folds around to a short-stacker NIT in the BB, which 3-bet jams all-in for $56. The action is on us. What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $63 Pot Odds Ratio = $63:$50 = 1.26:1 Pot Odds Pot Odds % Method = Call Size / Pot Size + Call Size Pot Size + Call = $63 Pot Size + $50 Call from You = $113 Pot Size % = $50 / $113 = 44% Pot Odds What is our opponent’s estimated jamming range? This is a very similar situation to our previous example, except now our short-stacker opponent is a NIT. Knowing that NITs are the scrooges of poker, we can expect him to have a very strong range that easily dominates ours. Pocket Pairs : JJ+ Broadway Cards : AK What is our estimated equity versus our opponent’s jamming range? Referring to the equity chart, we should expect to be a huge equity dog versus a NITs all-in jamming range with 77. We could reasonably estimate to have about 30% equity knowing we have a dominated range and aren’t the favorite to win the hand if we call. Again, we’ll utilize Equilab to provide a definitive estimate: Our Equity : 33% Opponent’s Equity : 67% Should we call? No. Getting 44% pot odds and only having 33% equity makes calling a –EV play. Since there are no implied odds in this situation, folding is our only option Playing a $1-$2 No-Limit Hold’em game, a relatively conservative TAG open-raises to $6 in the CO position. We look down at A♠ Q♠ on the BTN and 3-bet to $18. Both of the blinds fold, and the TAG 4-bet jams all-in for $125 total. With $200 behind, the action is on us What are the pot odds? Pot Odds = [pot size]:[amount to call] Pot Size = $146 Pot Odds Ratio = $146:$107 = 1.36:1 Pot Odds Pot Odds % Method = Call Size / Pot Size + Call Size Pot Size + Call = $146 Pot Size + $107 Call from You = $253 Pot Size % = $107 / $253 = 42% Pot Odds What is our opponent’s estimated jamming range? This is an interesting spot. We know that villain is a somewhat conservative TAG; however, a decent TAG will also know that we’ll potentially be 3-betting light on the BTN versus a wider CO opening range. Given this, we shouldn’t necessarily expect villain’s 4-bet jamming range to be ultra-tight. Instead, we should expect a much looser range, similar to the ultra-loose range discussed earlier in this chapter: AA, KK, QQ, JJ, TT, AK, AQ, and AJs. What is our estimated equity versus our opponent’s jamming range? While we estimate villain’s 4-bet jamming range to be fairly loose, our light 3-bet puts us in a precarious spot. Even against such a loose range, AQ is only ahead of AJs. Knowing this, our equity is very minimal in this situation. A conservative estimate would be around 30% equity. Again, we’ll utilize Equilab to provide a definitive estimate: Our Equity : 40% Opponent’s Equity : 60% Should we call? No. Getting 46% pot odds and only having 40% equity makes calling a –EV play. Since there are no implied odds in this situation, folding is our only option Playing a $1-$2 No-Limit Hold’em game, a solid LAG with a $440 stack open-raises to $6 on the BTN, a bad aggressive short-stacker with a $60 stack 3-bet jams all-in, and the action is on us in the BB. With a $320 stack, we look down at A♦ A♣. What is the optimal move in this situation, flat calling or cold 4-betting? What are our opponents’ estimated ranges? Given we are in a multi-way hand with multiple opponents, we must estimate both of our opponents’ ranges. First, we’ll tackle the LAG on the BTN. A solid loose aggressive opponent will be open-raising a very wide range on the BTN, attempting to steal our blinds. It wouldn’t be unreasonable to assign him a range as loose as a 50% opening range: BTN’s Estimated Open-Raising Range: 22+, A2s+, K2s+, Q4s+, J7s+, T8s+, 97s+, 86s+, 75s+, 64s+, 54s, A2o+, K7o+, Q8o+, J8o+, T8o+, 97o+, 87o, 76o, 65o, 54o Knowing the short-stacker in the SB is a bad aggressive maniac with a 30bb stack, we should also expect him to have a fairly wide jamming range, including all pocket pairs, all Broadway cards and some decent suited connectors: SB’s 3-Bet All-In Jamming Range: 22+, ATs+, KTs+, QTs+, JTs, T9s, 98s, 87s, ATo+, KTo+, QTo+, JTo Call or Raise? We know we’re an equity favorite in this hand; however, we must determine whether to simply flat call the raise or cold 4-bet re-raise. Let’s examine the merit of both. Merits of 4-Bet Re-Raising By 4-bet re-raising, we put immense pressure on BTN by showing a lot of strength. With 160bb effective stacks, we would expect him to fold a large portion of his hands and only continue with AA, KK, QQ, and AKs a majority of the time. Given that BTN is opening-raising a very wide range, we would expect him to fold greater than 95% of the time. By folding out BTN, we also ensure we maintain a higher equity edge versus SB, not having to give up some post-flop equity to BTN. This essentially ensures an all-in, heads up situation versus SB, where we’re a huge equity favorite to win the $68 already in the pot. Merits of Flat Calling By flat calling, we under-represent our hand, making it look deceptively weak to BTN. We represent hands such as 77, 88, 99, TT, AQs and AKo, where we’re willing to risk 30bb, but not go all-in. This encourages two potential actions from a loose aggressive BTN. First, assuming we don’t want to call an all-in with our 3-bet flatting range, BTN may be encouraged to 4-bet jam to scare us out of the pot with a semi-decent range of hands that do well against SB’s jamming range. Second, if BTN isn’t willing to 4-bet jam, flatting gives him a better price to call behind with a range of hands we dominate post-flop. With either of these two scenarios, we now have the ability to potentially win up to an additional $260 in implied odds money, given $320 effective stack sizes between us and BTN. Knowing this, calling is much more profitable than 4-bet re-raising in this spot. 4-Betting Potential Winnings : $68 in Current Pot Flat Calling Potential Winnings : $68 in Current Pot + $260 in Implied Odds Money Set-mining is when you call a pre-flop raise with the sole intention of flopping a set with a small pocket pair such as 22-55. Poker players love the thrill of flopping a set and stacking their opponents, but this doesn’t occur that often. As a consequence, lots of poker players set-mine incorrectly, therefore losing potential profit The odds of flopping a set or better as being 11.76% or 7.5:1, or simply, 1 in every 8.5 times Because we’ll only flop a set every 1 in 8.5 times we try, we have to make sure that during the one time we do, we make enough money to make up for the other 7.5 times we don’t flop. Since we’ll rarely get a good pot odds price to call pre-flop, set-mining is an implied odds play that relies on us making a decent profit post flop Since we lose 3bb every time we don’t flop a set, the one time we do, we need to win 24bb – not including rake – just to break-even. Since we’re not interested in break-even decisions, but rather, profitable ones, let’s move on to discuss profitable set-mining To ensure our set-mining decisions are profitable, we should follow the 15-to-1 rule. This rule states that for every 1bb we invest, we should expect a return of 15 times our initial 1bb investment, we would then need a (3bb x 15 =) 45bb return on our investment every time we flop a set Since set-mining relies on implied odds, we need to take several factors into consideration. First, we must consider the effective stack sizes in the hand to determine if we can get a 15 times return on our investment. Second, we need to consider the strength of our opponent’s hand. If it’s strong, we might have good implied odds. Conversely, if it’s weak, we won’t get good implied odds. Lastly, we need to consider our opponent’s playing style and if we expect him to pay us off or not when we hit our set A bad aggressive opponent with a 168bb stack open-raises to 4bb and the action folds around to us in the BB. We look down at 2♥ 2♣ with a 100bb stack. Should we call? Let’s examine our profitable set-mining criteria to determine if we can call or not. Effective Stack Sizes Basing our call upon the 15-to-1 rule, we need to win 60bb from villain if we hit our set, so with 100bb effective stack sizes, we meet this criterion. Opponent’s Hand Strength What about our opponent’s hand strength? Given that he is a bad aggressive opponent, we don’t expect him to have a great starting hand that often. Unfortunately, we don’t meet the second criterion. Implied Odds Do we expect to get paid when we hit our set? Given that our opponent is a bad aggressive maniac, he’ll often bluff post-flop and bet heavily with top pair. So, yes, we expect to get paid off nicely when we hit our set a decent amount of the time. Should We Call? While we don’t expect our opponent to have that great of a hand, we do stand the chance to win up to 100bb from our opponent, which is well above our 60bb requirement. We also expect him to spew off a lot of chips post flop whenever he flops a good pair or is entirely bluffing. So yes, we can call in this situation, given that our opponent is a bad aggressive maniac A conservative TAG with a 147bb stack open-raises to 3bb pre-flop from UTG, a loose passive opponent with a 47bb stack calls and the action is on us with 4♦ 4♣ with a 105bb stack. Should we call? Let’s examine our profitable set-mining criteria to determine if we can call or not. Effective Stack Sizes Basing our call upon the 15-to-1 rule, we need to win 45bb if we hit our set. With a 105bb effective stack size between us and the UTG TAG, we meet this criterion. Also, with a 47bb effective stack size between us and the loose passive opponent, we also meet this criterion. Opponent’s Hand Strength What about our opponents’ hand strengths? We expect the initial UTG raiser to have a relatively strong opening range. However, the loose passive caller will typically have a fairly weak calling range. Implied Odds Do we expect to get paid when we hit our set? It really depends. We shouldn’t expect either opponent to commit 45bb to the pot without an extremely strong hand. Typically, a conservative TAG won’t overplay and stack off with top pair alone. Our loose passive opponent will also most likely play in a fit-or-fold fashion since he’s not a calling station. Knowing this, we don’t have great implied odds in this situation. Should We Call? We probably shouldn’t call in this situation due to our implied odds not being that great. We want to set-mine in situations where we have a high probability of getting paid off and this just isn’t one of those situations We open-raise to 3bb with 3♥ 3♣, a NIT 3-bets to 10bb, and the action is back on us. Effective stack sizes are 89bb. Should we call? Let’s examine our profitable set-mining criteria to determine if we can call or not. Effective Stack Sizes Since this is a 3-bet pot, we need to call an additional 7bb to continue in the hand. Basing our call upon the 15-to-1 rule, we then need to win 7bb x 15 = 105bb to make this a profitable call. Since effective stack sizes are only 89bb, we don’t meet this criterion. However, with our break-even point being 8:1, we know that 7bb x 9 = 63bb is our breakeven point. So, while we can’t win 15 times our investment, we can win well over 9 times it – just under 13 times our investment, to be exact. Opponent’s Hand Strength What about our opponent’s hand strength? Given that our opponent is a NIT and has 3-bet us, we should put him on an extremely strong range that he is willing to stack off with post-flop on most board textures. A good estimate would be AA, KK, QQ and AKs. Implied Odds Do we expect to get paid when we hit our set? Most definitely. NIT’s only 3-bet is with the very top portion of their range, which they’re willing to stack off with in most situations. So by flat calling, we can expect to get stacks all-in in many situations post-flop when we hit our set. Furthermore, it’s always easier to get stacks all-in in 3-bet pots, given the larger pot sizes going to the flop. Should We Call? Yes, we can definitely call in this situation based upon several factors. While we aren’t getting 15-to-1 on our money, we are getting close to 13-to-1 if we stack our opponent, which is still profitable for set-mining purposes. Additionally, we expect our opponent to have an extremely strong range that will stack off post-flop in most situations Playing from the BTN is the most profitable position in No-Limit Hold’em because it allows us to steal our opponents’ blinds with a wide range of cards. Theoretically, we should be opening a very wide range of hands from late position in an attempt to steal the blinds When we’re attempting to steal our opponents’ blinds, we’re risking a certain amount to win a certain amount. Typical blind steal sizing is 2bb, 2.5bb or 3bb. In most games, the small blind will be 0.5bb and the big blind with be 1bb. So we’re typically risking 2bb – 3bb to win 1.5bb Blind Steal Break Even Percentage : Risk / (Risk + Reward) 3bb Steal Sizing : 3bb / (3bb + 1.5bb) = 66.7% Breakeven Point 2.5bb Steal Sizing : 2.5bb / (2.5bb + 1.5bb) = 62.5% Breakeven Point 2bb Steal Sizing : 2bb / (2bb + 1.5bb) = 57.1% Breakeven Point What this shows us is that, on the average, our blind steal attempts need to work approximately two-thirds of the time. Furthermore, a 2bb sizing needs to work approximately 10% less often than a 3bb sizing. In general, you want to risk the least amount possible when stealing your opponents’ blinds We want to steal our opponent’s blinds with our mediocre starting hands when we expect them to fold more often than our breakeven point When our opponents don’t fold, we need to take potential pre-flop and post-flop actions and scenarios into consideration. Our Opponents’ Overall Playing Style : We need to take our opponents’ overall playing style into consideration, as well as whether our opponents are good or not. Different types of players will defend their blinds at different frequencies and also play differently post-flop. Good players will make fewer mistakes while bad ones will make many mistakes both pre- and post-flop. Our Opponents’ Blind Defense Propensity : Certain opponents will fold a very high frequency of hands in the blinds, whereas others will not. We can attack opponents folding a very high frequency with a wide stealing range, but we need to steal more cautiously versus those who defend their blinds more liberally by calling or 3-betting as a bluff. Our Opponents’ Post-Flop Tendencies : Stealing the blinds from fit-or-fold opponents is much easier than stealing from good opponents that will defend well from the blinds, but we need to be on the lookout for both types of opponents. Moreover, we should target bad fit-or-fold opponents and be wary of stealing against better opponents. Understanding these considerations is vitally important because our steal attempts aren’t played in a vacuum. Whenever we’re considering a steal attempt, we need to take all pre- and post-flop scenarios and our opponents’ playing styles into consideration The action folds around to us on the BTN with A♣ 3♥ with two NITs in the blinds. Should we attempt to steal their blinds? A3o is by no means a premium starting hand, so it’s not a hand we want to be stealing with too lightly against the wrong opponents. Luckily for us, both of our opponents in the blinds are NITs, who characteristically have a tendency to fold too much pre-flop due to their risk-adverse nature. Given this information, we should definitely attempt to steal their blinds with a 2-2.5bb steal sizing The action folds around to us on the BTN with K♣ T♥ with a NIT in the SB and a loose passive calling station in the BB. Should we attempt to steal their blinds? KTo, while it appears to be a good starting hand, is a mediocre Broadway hand. On the other hand, this is a good situation to steal for two main reasons; first, we expect the NIT in the SB to fold a high percentage of the time. Second, and most importantly, we also expect the loose passive calling station in the BB to defend his blinds very liberally, with a wide range of hands. Our goal with this steal attempt is not to make BB fold, but instead call so we can make money off of his post-flop mistakes. Luckily, KTo has good equity against a loose passive calling station’s calling range, providing us with the opportunity to win additional money post-flop whenever we flop top pair or better – simply because calling stations hate to fold. When we miss, King-high will often be good at showdown. Knowing all of this, we should be attempting to steal our opponents’ blinds with a 3bb steal sizing The action folds around to us on the BTN with J♣ 7♣ with a TAG in the SB and a LAG in the BB. Should we attempt to steal their blinds? This is a very unfavorable steal situation due to both of our opponents in the blinds. First, both are good, aggressive opponents that won’t make many mistakes both pre- and post-flop. Second, since they’ll both be aware that we’ll be attempting to steal their blinds lightly, we can expect them to defend their blinds by 3-bet bluffing at a decent frequency to discourage us from doing so. Knowing this, we should be tightening up our stealing range against these types of opponents and simply throw our J7s into the muck We 3-bet for two specific reasons: When we have a hand that’s too good to call, such as KK or AA for value. When we have a hand that’s too bad to call, such as A2s or 33. Typically, 3-bet sizing is 3 times our opponent’s initial open-raise sizing, with some slight deviations depending upon if we’re in or out of position: Out of Position Sizing : When we’re out of position to the raiser, we should make our sizing a bit larger, closer to 3.5x due to our positional disadvantage in the hand. In Position Sizing : When we’re in position to the raiser, we should make our sizing a bit less, closer to 2.8x due to our positional advantage in the hand. Now that you know the basics of 3-betting, as well as common 3-bet sizings, let’s calculate 3-bet bluff breakeven points. For these calculations, we’ll assume our opponent has open-raised to 3bb and there is 1.5bb in the pot from the SB and BB. 3-Bet Bluff Break Even Percentage : Risk / (Risk + Reward) 2.8x Sizing : 8.4bb / (8.4bb + 4.5bb) = 65.1% Breakeven Point 3x Sizing : 9bb / (9bb + 4.5bb) = 66.7% Breakeven Point 3.5x Sizing : 10.5bb / (10.5bb + 4.5bb) = 70% Breakeven Point What you’ll notice is that, on the average, our 3-bet bluff attempts need to work around two-thirds of the time, with larger bluffs having to work a bit more often. All-in-all, 3-bet bluffs are a risky endeavor because we are re-raising the initial raiser, risking approximately 9bb pre-flop, so we need to win approximately 67% of the time just to break even. On top of that, if our opponent calls, we stand to lose additional money post-flop Because 3-bet bluffing is a risky endeavor, we should have a very good reason for attempting this play. There are two primary reasons why we would want to attempt this play: Balancing Our Ranges : When we’re balancing our ranges, we’re 3-bet bluffing to ensure we get paid off when we value 3-bet. High Fold Equity : The second instance that we would attempt a 3-bet bluff in is when we assume we have high fold equity, meaning we expect our opponent to fold a high percentage of the time to our 3-bets. This typically would occur when our opponents are opening a wider, weaker range. Therefore, we should only apply the polarized 3-betting model with 3-bet bluffs if our opponent(s) are folding to a lot of 3-bets. If they aren’t, then 3-bet bluffing will only cause us to unnecessarily spew off a lot of chips pre-flop. Conversely, if our opponent(s) are folding to a high frequency of 3-bets, approximately 67% at minimum, then we can 3-bet bluff profitably. Therefore, 3-bet bluffing and the polarized 3-betting model works best when we have a lot of fold equity. When our fold equity is low, we shouldn’t apply this model. Instead, we should stick with the linear 3-betting model A NIT open-raises to 3bb in UTG and the action is on us. We look down at 8♣ 7♣. Would this be a good time to consider a 3-bet bluff? Hopefully, you’ve already come to the correct conclusion with this hand. When a NIT open-raises from UTG, he has a very strong starting range; therefore, we should expect to have very little fold equity. Knowing this, we shouldn’t attempt a 3-bet bluff in this situation A TAG open-raises to 3bb from the BTN as a steal attempt, SB calls, and the action is on us with 2♦ 2♥. The TAG on the BTN has been attempting to steal our blinds every single orbit, so we know he is opening a wide range. Would this be a good time to consider a 3-bet bluff? This is a great spot to consider a 3-bet bluff, as well as consider a re-steal in this specific situation. Since the TAG on the BTN has been attempting to steal our blinds every single orbit, we know he is opening a very wide range with a lot of mediocre hands that cannot profitably call a 3-bet. We also have a re-steal situation, meaning we can re-steal SB’s call, which is typically composed of a weak, capped range that doesn’t include hands strong enough to call a 3-bet. A LAG open-raises to 3bb from MP, a loose passive calling station in the CO calls and the action is on us on the BTN with A♠ 4♠. Would this be a good time to consider a 3-bet bluff? While we know that a LAG will be opening a fairly wide range, we also know that he’ll be likely to fight back in a smart and aggressive way against our 3-bet attempts. Additionally, the loose passive calling station will be apt to call our 3-bet as well. Even though both of our opponents most likely have weaker starting ranges, we shouldn’t expect both of them to fold that often. Knowing this, we shouldn’t attempt a 3-bet bluff in this situation Board texture plays an incredibly important role in determining the likelihood that our opponents have a drawing hand or not, as well as how much we should bet in certain situations. There are two specific types of board textures: wet and dry Wet board textures are draw-heavy, coordinated board textures with the likelihood of numerous possible draws; the most important among them being flush and straight draws. Below are two sample wet boards: Example #1 : A♣ J♣ T♣ Example #2 : 5♦ 6♦ J♥ As you can see with both examples, there are numerous possible draws. Example #1 is an example of an extremely wet board texture, while Example #2 is an example of a semi-wet board texture Because wet boards increase the likelihood of our opponents having strong drawing hands, we often need to bet for both value and protection The opposite of wet board textures are dry board textures. Dry board textures are non-coordinated boards with little to no possible draws. On a dry board texture, our opponents either have a made hand or not, with very little chance of making a drawing hand by the river. Typically with dry boards, our opponents will potentially have only a backdoor draw, meaning they need two specific cards to make their draw on the river. If both cards don’t come, they won’t make their draw. Below are two sample dry boards: Example #1 : 2♦ 6♥ J♣ Example #2 : K♦ 8♠ 2♥ As you can see, there are very few possible draws in either hand shown above. In fact, there are only backdoor draws. The only way our opponent(s) can make a flush or straight is with runner-runner turn and river cards on the river. Therefore, we’re less worried about protecting our value hands on dry boards, since there aren’t many drawing hands that can suck out on us Our bet sizing will be based upon our opponents’ perceived equity Knowing how to quickly determine what draws our opponents most likely have based upon the board texture combined with each draw’s equity is very important. We use our opponents’ estimated equity to determine our proper value bet sizing. The goal is to always give our opponents a bad pot odds price to call. We should also be seeking to extract maximum value as well Whenever we bet, we’re offering our opponents a gambling wager. our main goal with betting is to provide our opponents a –EV pot odds price, forcing them to call more than they should. Therefore, whenever we bet, we should always consider the pot odds price we’re offering our opponent(s) based upon their estimated equity to hit their draws < This table shows common bet sizes and their associated pot odds. For example, a 1/2 pot-sized bet equates to offering our opponent(s) 25% pot odds. > While the table highlights recommended minimum bet sizing, we should always consider the maximum bet sizing we think our opponent(s) will call based upon their playing style and tendencies. For example, if our opponent is a calling station, we should definitely bet more; however, if our opponent is a NIT, we should bet the recommended minimum amount instead. All-in-all, if we think our opponent will call a bigger bet size, by all means we should bet more. At the lower stakes and micro stakes levels, most opponents will call a large bet with a flush or straight draw. Most are simply unaware of the basic math behind their draws and will often make –EV calls, so regardless of their draw, we can get away with betting more against bad, recreational players There are certain situations in which over-betting the pot is correct. This is when we think our opponents could have a flush + open-ended straight draw with 15 outs and 33% equity. Whenever this is the case, we need to slightly over-bet the pot to ensure we offer them a –EV pot odds call. The reason for this is that a pot-sized bet equates to 33% pot odds. If we estimate our opponent to have 33% equity, we need to offer them more than 33% pot odds – hence the over-bet. My recommendation is to slightly over-bet the pot to around a 1.2 to 1.3 pot-sized bet. However, you’ll often find that when someone has such a strong draw, he’ll often re-raise all-in on the flop, because with 15 outs, such a strong draw is greater than a 50% equity favorite to win all-in by the river There are certain situations in which over-betting the pot is correct. This is when we think our opponents could have a flush + open-ended straight draw with 15 outs and 33% equity. Whenever this is the case, we need to slightly over-bet the pot to ensure we offer them a –EV pot odds call. The reason for this is that a pot-sized bet equates to 33% pot odds. If we estimate our opponent to have 33% equity, we need to offer them more than 33% pot odds – hence the over-bet. My recommendation is to slightly over-bet the pot to around a 1.2 to 1.3 pot-sized bet. However, you’ll often find that when someone has such a strong draw, he’ll often re-raise all-in on the flop, because with 15 outs, such a strong draw is greater than a 50% equity favorite to win all-in by the river We have A♣ K♣ and the flop is K♥ 9♥ 4♠. What’s the board texture? This is a semi-wet board texture, with both a flush and gut-shot straight draw. What is our opponent’s drawing hand estimated equity? A flush draw and gut-shot straight draw have a combined 25% equity if we suspect our opponent can have both draws. How much should we bet? We should bet at minimum a 2/3 pot-sized bet, which will offer 28.5% or more pot odds. If we think our opponents will call a 3/4 or pot-sized bet, we should bet more to extract additional value. But at minimum, we should bet a 2/3 pot-sized bet We have 7♦ 7♣ and the flop is 5♥ 6♥ 7♠. What’s the board texture? This is a very wet board texture with both a flush and open-ended straight draw. Most likely, we have the best hand with the top set however; it is possible one of our opponents could have flopped the straight. What is our opponent’s estimated drawing hand equity? Such strong draws combined can have up to 33% equity in the hand, so we need to bet big to give our opponents a bad price to call. How much should we bet? We should over-bet the pot to around a 1.2 to 1.3 pot-sized bet, and if we are re-raised all-in, we should definitely consider calling with top set. Even if we are behind, we have redraws to quads or the full house We have K♣ Q♦ and the flop is K♥ 7♣ 2♦. What’s the board texture? This is an extremely dry flop. We almost certainly have the best hand, unless our opponent flopped two pair or a set, which are both unlikely. What is our opponent’s estimated drawing hand equity? Our opponent either has a made hand, or doesn’t, without any draws on the turn. We’re either way ahead or way behind in this spot to two pair or sets. How much should we bet? On such a dry board texture, we should bet no more than a 1/2 pot-sized bet. If we think our opponents are unlikely to have a King, then we should also consider betting less or checking the flop to allow our opponents to catch up on the turn or induce a bluff. The benefit of checking here is two-fold. Firstly, it induces our opponents to potentially bluff into us by sensing weakness with our check. Secondly, it allows our opponents to catch up by making a pair on the turn, or to think a pair of 7’s, or small pocket pairs such as 88 or 55, are potentially good on the flop. If we’re in position, we also allow our opponents to bet the turn with smaller pairs, or call our delayed turn continuation bet with weaker hands, thinking their smaller pairs are potentially the best hand We open raise to 3bb UTG with A♦ Q♦, a loose passive calling station in the CO calls, and everyone else folds. Effective stack sizes are 115bb going to the flop. The flop is A♣ Q♣ T♣. What’s the board texture? This is an extremely wet flop with numerous flush and straight draws, as well as possible made flushes and nut straight. Our opponent can make a flush with a single club card and the nut straight with either a King or Jack. What is our opponent’s estimated drawing hand equity? We know that our opponent is a calling station. Therefore, we should estimate that his pre-flop calling range is too wide, and includes lots of mediocre hands. This is usually a good thing; however, in this situation, having such a wide pre-flop calling range allows our opponent to easily have a random club card flush draw and numerous weaker Kx and Jx hands for straight draws. Without a doubt, we should assume our opponent could have a combination of a flush + gut shot straight draw giving him 25% equity. How much should we bet? At minimum, we should bet a 2/3 pot-sized bet to give our opponent 28.5% pot odds. But, understanding that he is a calling station, we can easily get away with betting more. Against an opponent that isn’t folding his draws, we should be looking to extract maximum value. Accordingly, this would be an opportune spot to over bet the pot to around a 1.3 pot-sized bet. We open raise to 3bb in MP with A♠ A♦, an aggressive LAG calls from the CO and everyone else folds. Effective stack sizes are 178bb going to the flop. The flop is 2♥ 7♣ Q♠ and the action is on us. What’s the board texture? This is a very dry board texture with only backdoor runner-runner draws. We can safely assume we have the best hand the majority of the time, unless our opponent has flopped a set. What is our opponent’s estimated drawing hand equity? We know that our opponent is an aggressive LAG. Therefore, we should estimate that his pre-flop calling range is somewhat wide, but shouldn’t include too many junk hands. We would expect him to call with Broadway cards not good enough to 3-bet, suited connectors, suited one-gappers, and medium to small pocket pairs. It’s possible our opponent also has a pair with Qx hands, 7x hands, and pocket pairs. He will also flop a set of 2’s or 7’s a very small percentage of the time. We wouldn’t expect him to have QQ, since we would expect him to 3-bet such a strong hand pre-flop. Knowing this, we almost always have the best hand in this situation. Furthermore, our opponent has very little equity to make the best hand by the river with a runner-runner flush, straight, three of a kind, or two-pair draw. How much should we bet? We almost always have the best hand, and we’re facing a good, aggressive opponent out of position. Since we aren’t concerned with betting for protection, our sole mission with this hand is to extract as much value as possible from our opponent. Value-betting the flop around a 1/2 to 2/3 pot-sized bet is a good option here, because we don’t expect our opponent to fold a pair of Q’s, 7’s or 88-JJ. Checking to induce a bluff is another great option. If we expect our opponent to value bet worse hands or bluff when we check the flop first to act, we can extract value by showing weakness on the flop. All-in-all, the most optimal move depends on our previous reads on our opponent – but both are good options A semi-bluff occurs when you bet with a drawing hand, such as a flush draw, straight draw, or over cards. While it’s a bluff at the time of your bet, you stand a chance of making your draw on the next street of action to make a better hand than your opponent, hence it becomes a semi-bluff. A pure bluff, on the other hand, has very few to no draws at all When we semi-bluff with the intent of going all-in, we’re concerned with our all-in equity. Reviewing the handy Rule of 2 and 4, we multiply our outs by either 4 or 2, depending on whether we are going all-in on the flop or the turn to determine our all-in equity: Use the Rule of 2 and 4 to find your all-in equity: On the Flop: multiply your outs x 4 On the Turn: multiply your odds x 2 In a heads up hand, if we have approximately 50% equity, we are happy to semi-bluff all-in on the flop. In a 3-way hand, we can optimally go all-in with around 33% equity, if we expect both opponents to call When we semi-bluff, whether it’s an all-in situation or not, we have two primary goals. Our first goal is to maximize fold equity by forcing our opponent into a tough decision, hoping he folds. When we’re (semi) bluffing, we’re always happy when our opponent folds and we can take down the pot right then and there without going to showdown, simply because we’re betting with the worst hand. Our secondary goal with semi-bluffing is to maximize our value when we hit our hand. When we bet or raise all-in with a semi-bluff, we ensure that we maximize our value when we’re called and make our hand. This is why you’ll often see people aggressively semi-bluffing the nut flush all-in; if they simply took a check/call line, when the flush hits most opponents, it will shut down and not pay them off since it’s an obvious draw. All-in-all, we’re happy if our opponents either fold or call our all-in semi-bluffs We have A♣ K♣ and the flop is Q♣ J♣ 4♠. Villain bets $75 into a $100 pot, making it $175, and the action is on us. Is this a good semi-bluff spot? Pot Odds: Given the bet size, we are being offered 2.3:1 pot odds, which converts to 30% pot odds. Outs : We have 9 outs to make the nut flush draw, plus an additional 3 outs to make the nut straight, giving us a total of 12 outs. Not All-In Equity : 12 outs x 2 = ~24% Equity All-In Equity : 12 outs x 4 = ~48% Equity Calling would be slightly –EV, since we are being offered 30% pot odds with a 24% equity chance of making our draw on the turn. If either the flush or straight completes on the turn, our opponent might shut down and not pay us off, rendering our implied odds minimal. With that being said, this is a great spot to semi-bluff. If we think our opponent will fold hands, such as top pair to a flop check/raise all-in, this is an ideal spot to maximize our fold equity. Furthermore, with approximately 48% flop all-in equity, we’re okay with being called, as this is essentially a coin-flip scenario We have 8♦ 9♦ and the flop is 6♣ 7♥ K♠. Villain bets $50 into a $100 pot, making it $150, and the action is on us. Is this a good semi-bluff spot? Pot Odds : Given the bet sizing, we’re getting 3:1 pot odds, which converts to 25% pot odds. Outs : We have 8 outs to make the straight with our open-ended straight draw. Not All-In Equity : 8 outs x 2 = ~16% Equity All-In Equity : 8 outs x 4 = ~32% Equity This is a very interesting spot with great implied odds. In a heads up pot, it’s not an ideal semi-bluff spot, based upon our equity. If it were a multi-way pot with several aggressive opponents, then given our approximately 32% flop all-in equity, semi-bluffing all-in is potentially correct, given the math. However, given that this is a heads up situation, calling is much more preferable than semi-bluffing, which I’ll explain below. When our opponent continuation bets into us, it’s very likely that he has a pair of Kings. Knowing this, even though we are getting poor direct pot odds, we have great implied odds with a hidden draw. Neither card that completes our straight will deter our opponent from continuing with a pair of Kings; if the turn or river card is a T or 5, we wouldn’t expect our opponent to shut down. This ensures that we get implied odds value from our hand A NIT open-raises to 4bb UTG, we call with K♥ Q♥ from MP, CO folds, BTN calls, and both blinds fold. The flop is J♥ T♥ 2♣, UTG continuation bets a 10bb into a 13.5bb pot and the action is on us with 110bb effective stack sizes. Is this a good semi-bluff spot? Pot Odds : Given the bet sizing, we’re getting 2.35:1 pot odds, which converts to 30% pot odds. Outs : We have 9 outs to make the flush and an additional 6 to make the straight, giving us a total of 15 outs. Not All-In Equity : 15 outs x 2 = ~30% Equity All-In Equity : 15 outs x 4 = ~60% Equity (54.1% Actual Equity) This is a dream situation when it comes to semi-bluffing our hand. We have a royal flush draw combined with an open-ended straight draw, giving us more than enough equity to semi-bluff raise with the intention of going all-in on the flop. Yes, we’re getting the correct pot odds to flat call. However, with such a strong hand (a 54% favorite) to make the best hand by the river, raising makes much more sense in this specific situation. When a NIT open-raises to 4bb UTG and continuation bets very large amounts into two opponents on this flop, we would expect him to have a very strong made hand, possibly as strong as a set of J’s or T’s. This increases the likelihood that he’ll call our semi-bluff raise or re-raise us all-in. However, if a heart or a card that completes the straight comes on the turn or river, he may stop being a cautious NIT. For this reason, semi-bluffing with the intention of going all-in on the flop is preferable to simply calling with this hand A hero call is simply when you call a potential bluff on the river with a marginal hand, hoping to catch your opponent bluffing. It’s called a hero call because you’re the hero in the hand when you catch your opponent bluffing, and the table is in awe of your opponent reading abilities! For example, you have 8♣ 8♥ and call an aggressive opponent’s pre-flop raise. The flop comes 2♦ 4♦ 9♣. Your opponent fires out a continuation bet and you call. The turn comes T♠. Your opponent bets again and you call. The river comes 7♣. You opponent goes all-in. You think about it for a few minutes, and decide to call. Your opponent turns up A♦ K♦ and you take down a massive pot. This would be considered a hero call because you called your opponent’s bluff with a mediocre hand, which in this case was the third best pair When we’re bluffing, we’re risking our bet-size to win the amount of money that’s already in the pot. Therefore, we have a simple risk-to-reward ratio where we risk a certain amount to win a certain amount: Risk : Our Bet Size Reward : The Pot The pot is $100 and we bluff $50. How often does our bluff need to work? Reward:Risk Ratio = $100:$50 = 2:1 = 1/3 = 33.3% In this situation, we’re risking $50 to win the $100 already in the pot. You can see from the simple ratio above that our break-even point is 33.3%. Therefore, our bluff needs to work 33.3% of the time to be break-even. So, if it works more than 33.3% of the time, it’s profitable! The math is as simple as that; but there are many more factors that go into successfully bluffing such as the board texture, our perceived ranges, effective stack depth, our opponents’ playing styles, the stakes we’re playing, and so forth Hero calls, and how often they need to work, are based purely upon the pot odds we’re being offered. If we’re being offered 25% pot odds, then our hero calls need to be right more than 25% of the time to be profitable Just like bluffs, there is a lot that goes into making profitable hero calls. We really need to understand how our opponent is playing in order to determine if he is bluffing or not on the river. Learning to read our opponent’s ranges and how they connect with the board texture, as well as our opponent’s propensity to bluff rivers will help improve the overall success of your hero calls. Remember, the math discussed above won’t tell us if a hero call will work, just how often it needs to work The basic EV calculation is very simple and is composed of two parts: EV = [Part A] - [Part B] Part A : How often you win x How much you win Part B : How often you lose x How much you lose EV = [Expected Long-Term Winnings] – [Expected Long-Term Loses] Perform a basic EV calculation using the 3-step methodology shown below: Step 1 : Determine how often you will win and how much you will win:   80% of the time you will win $500 = (.80 x 500) = $400   Step 2 : Determine how often you will lose and how much you will lose:   20% of the time you will lose $250 = (.2 x 250) = $50   Step 3 : Subtract how much you expect to lose from how much you expect to win: $400 - $50 = $350 +EV This play is +EV and you expect to win $350, on the average, in the long run Pre-flop, we have J♣ J♠ and villain open-jams all-in with $80 effective stack sizes in a $500 buy-in deep-stack game. We estimate that we’re a 59% favorite to win and will lose 41% of the time, based upon our opponent’s 3-betting frequency. If we call and win, we’ll win a total of $161.50, which includes the blinds. However, if we call and lose, we’ll lose the $80 we risked pre-flop by calling villain’s all-in jam. What is the EV calculation for this play? EV = [Part A] - [Part B] Part A : How often you win x How much you win Part B : How often you lose x How much you lose Part A : How much we will win x percentage to win = ($161.50 x .59) = $95.29 Part B : How much we will lose x percentage to lose = ($80 x .41) = $32.80 EV = $95.29 - $32.80 = $62.49 Is it a +EV or -EV play? This is a +EV play. Each time you make this play, you can expect to win $62.49 on average, over the long run We raise UTG to $6 with A♣ A♦ and get called by one opponent. The flop comes K♠ 8♥ 3♠. We bet $12 into a $15 pot, our opponent raises to $40 and we go all-in for our remaining $180 stack. Having us covered, our opponent calls and turns up K♣ 9♦. The turn card is 9♥ and the river card is Q♣, causing us to become unlucky and lose a massive pot. While we lost this particular hand, as well as our entire stack, we made a very profitable long-term winning play. We’ll do a simple EV calculation to show exactly how profitable our play was. Determining Win & Loss Information We first need to determine how often we expect to win. Using Equilab, we determine that we expect to win and lose: Win : 82.40% Lose : 17.60% Next, we need to determine how much we expect to win and lose with this play. If we win, we win a total amount of $399: $15 Flop Starting Pot Size $12 Bet from Us $40 Raise from Villain $180 All-In Jam from Us $152 All-In Call from Villain Total Pot Size : $399 If we lose, we lose our final $180 all-in bet, since the money already wagered by us in the pot no longer belongs to us. This is a very important concept to grasp, because we just lose our flop all-in re-raise amount of $180, not the entire $198 we committed to the pot. Win Amount : $399 Loss Amount : $180 What is the EV calculation for this play? EV = [Part A] - [Part B] Part A : How often you win x How much you win Part B : How often you lose x How much you lose Part A : How much we will win x percentage to win = ($399 x .824) = $322.78 Part B : How much we will lose x percentage to lose = (-$180 x .176) = $31.68 EV = (.824 x $399) - (.176 x -$180) = $291.10 EV = $322.78 - $31.68 = $291.10 Even though we lost a lot of money with this particular hand in a vacuum, in the long-run we expect to win $322.78 on average, making this an extremely +EV play. Nobody likes losing a lot of money at the poker table, but our EV calculation confirms we made the right play and that we shouldn’t dwell on the fact that we lost this particular hand. We should, in fact, be happy that our opponent called our all-in with such a weak hand and hope he continues to do so in the future, since we’ll win a lot of money from him in the long run. The key take away from this EV calculation is that poker is a game of long-term EV, not short-term wins and losses. Anybody can get lucky and win a massive pot, as well as get unlucky and lose a massive pot, in any single hand. Long-term results are what determine how profitable a poker player’s plays are Calculating EV in semi bluffs: When we either semi-bluff or purely bluff, we give ourselves two ways of winning a hand. The first is by making our opponent fold immediately, while the second is making the best hand at showdown when we do get called. This EV calculation takes both of those scenarios into account, making it a bit more complex than the basic EV calculation: Scenario 1 : Hero (semi) bluffs, villain folds, and hero wins Scenario 2 : Hero (semi) bluffs, villain calls and hero wins Scenario 3 : Hero (semi) bluffs, villain calls and hero loses Our EV calculation then becomes the following: Semi-Bluff EV = (Scenario 1) + (Scenario 2) – (Scenario 3) Scenario 1 = (Villain Fold % x Size of Pot) Scenario 2 = [(Villain Call % x Probability of Winning) x (Size of Pot + Villain Call)] Scenario 3 = [(Villain Call % x Probability of Losing) x (Semi-Bluff Amount)] We flop an open-ended straight draw with an estimated 8 outs and 32% all-in equity. We put our opponent on a very strong range and assume we have to make our draw to win the hand. There is $50 in the pot and our opponent bets $25. With $100 left in our stack, we go all-in as a semi-bluff. What is the EV calculation for this play? Semi-Bluff EV = (Scenario 1) + (Scenario 2) – (Scenario 3) Scenario 1 = (Villain Fold % x Size of Pot) Scenario 2 = [(Villain Call % x Probability of Winning) x (Size of Pot + Villain Call)] Scenario 3 = [(Villain Call % x Probability of Losing) x (Semi-Bluff Amount)] Estimating how often villain folds will always be an educated guess. You’ll never know exactly how often he’ll fold, so you should do your best to estimate based upon your opponent’s hand strength and playing style. In this example, we assume our opponent has a very strong holding and will only fold 20% of the time and call the remaining 80%. We also assume that if our opponent does call, we will only win 32% of the time. Semi-Bluff EV = (Scenario 1) + (Scenario 2) – (Scenario 3) Scenario 1 = (20% x $175) = $35 Scenario 2 = [(80% x 32%) x ($175 + $75)] = $64 Scenario 3 = [(80% x 68%) x ($100)] = $54.40 Semi-Bluff EV = $35 + $64 - $54.40 = $44.60 Is it a +EV or -EV play? This is a +EV play, if our assumptions are correct. When we semi-bluff, we must estimate how often our opponent will fold, i.e., how much fold equity we have in the hand. Sometimes we’ll be correct, while other times we’ll be incorrect. However, what this EV calculation shows is that, by semi-bluffing, we provide ourselves two ways to win the hand. If we correctly assume our opponent will fold 20% of the time, we’ll win $35 outright when he folds and $64 when he calls and we suck out to make the best hand. One thing to note with this calculation is that when we lose, we don’t take into account the total amount of money we invested into the hand, because the money in the pot no longer belongs to us. When we lose, we’re assuming we only lose our semi-bluff amount A LAG on the BTN open-raises to 3bb and it folds to us in the BB. With A♣ 5♣, knowing our opponent is attempting to steal our blinds with a wide range of hands, we 3-bet bluff to 10bb. Villain thinks for a minute, 4-bets to 24bb and we 5-bet shove with our remaining 90bb stack. What is the EV calculation for this play? Semi-Bluff EV = (Scenario 1) + (Scenario 2) – (Scenario 3) Scenario 1 = (Villain Fold % x Size of Pot) Scenario 2 = [(Villain Call % x Probability of Winning) x (Size of Pot + Villain Call)] Scenario 3 = [(Villain Call % x Probability of Losing) x (Semi-Bluff Amount)] This is an interesting spot that you’ll see often at the mid-to-higher stakes, where Meta game and balanced plays assume a major role in profitable poker. When we 3-bet our opponent, he knows we’ll sometimes defend our blinds with a 3-bet bluff, so it’s not unlikely that he will also sometimes 4-bet bluff us. For the sake of this example, we’ll assume he’ll fold to our 5-bet shove 40% of the time, however, when he calls, we only expect to win 20% of the time against his 5-bet all-in calling range. Semi-Bluff EV = (Scenario 1) + (Scenario 2) – (Scenario 3) Scenario 1 = (40% x 124.50bb) = 49.80bb Scenario 2 = [(60% x 20%) x (124.50bb + 76bb)] = 24.06bb Scenario 3 = [(60% x 80%) x (90bb)] = 43.20bb 5-Bet All-In Bluff EV = 49.80bb + 24.06bb - 43.20bb = 30.66bb Is it a +EV or -EV play? According to our sample analysis of our opponent’s likelihood of folding and our equity against his calling range, this is a +EV play Combinatorics Any 2 Suited Non-Pair Cards: 4 Combinations Any 2 Non-Suited Non-Pair Cards: 12 Combinations Any pocket pair can be combined 6 different possible ways, and any non-pair starting hand can be combined 16 possible ways, with 4 combinations being suited, and the remaining 12 being off-suit combinations A NIT open-raises to 4bb UTG and we have A♣ Q♣. How many combinations of pocket Aces does our opponent have in his UTG opening range? Using the simple 3 + 2 + 1 rule, we would eliminate “3” from our simple formula: 3 + 2 + 1 = 3 Combinations Left: Eliminated Combos : A♣ A♠, A♣ A♠, and A♣ A♦ Remaining Combos : A♦ A♥, A♠ A♦, and A♠ A♥ Every time we have an Ace or see an Ace on the flop, we eliminate one number, starting with the “3” going left to right. Accordingly, the 3 + 2 +1 rule tells us that our opponent can only have 50% of the total 6 combinations of pocket Aces when we hold an Ace in our starting hand Let’s take this example a bit further. Still assuming we have A♣ Q♣, and the flop is A♦ 7♠ 2♥, how many combinations of pocket Aces can our opponent have in his UTG opening range? Since we hold an Ace and there is a visible Ace on the flop, we would eliminate the “3” and “2” from the 3 + 2 + 1 rule formula: 3 + 2 + 1 = 1 Combinations Left: Eliminated Combos : A♣ A♠, A♣ A♠, A♣ A♦, A♦ A♥, and A♠ A♦ Remaining Combo: A♠ A♥ When there are two visible Aces, such as when we have one in our hand and one on the flop, there is only one remaining combination of pocket Aces available for our opponent to have in his UTG opening range. This reduces the likelihood that he has pocket Aces by 83.3% This concept is called blockers and card removal. In our previous examples, by having an Ace in our hand or an Ace on the board, we block our opponent from having certain combinations of pocket Aces. With specific cards removed from the deck, such as the A♣, our opponent cannot have any combinations of AA that include the A♣. Whenever potential cards that make up our opponent’s made or starting hands are no longer available to our opponent by either: (1) being in our starting hand, or (2) being on the board, we can use combinatorics to quantitatively remove combinations of hands from our opponent’s possible made hands in a process called card removal For any non-paired hand, such as AK, JT, or 56, there are a total of 16 combinations with 4 combinations being suited and the remaining 12 being unsuited. There is a simple calculation to show this. We will use AK as an example. There are 4 suits for any card (A♣, A♦, A♥, and A♠). So for AK, we know there are 4 Aces and 4 Kings in the deck. To find out the total possible combinations of AK, we would simply multiply: 4 Aces x 4 Kings = 16 Total Combinations We can further break down our 16 combinations to suited hands and off-suited hands:   4 Suited Combinations (A♣ K♣, A♦ K♦, A♥ K♥, and A♠ K♠) The remaining 12 combinations of AK are off-suited AK hands Estimating the number of combinations of any non-suited hand is really easy. We start with our simple 4 x 4 calculation, and whenever a card is taken via card removal, we reduce the number of cards left in the calculation. For example, if we continue with AK and an Ace hits the flop, there are 12 combinations of AK possible: 3 Remaining Aces x 4 Kings = 12 Combinations If the flop is K♣ J♣ T♥, how many combinations are there of KJ left in the deck? Well, we must eliminate the K♣ and J♣ from our calculation. So there are only 3 Kings and 3 Jacks left in the deck, giving us: 3 Kings x 3 Jacks = 9 Combinations If the flop is 9♣ T♣ T♥, how many combos are there of AT left in the deck? Well, we must eliminate two Tens from our calculation. So, there are two Tens left in the deck and 4 Aces: 4 Aces x 2 Tens = 8 Combinations We can use Combinatorics to help us with our difficult pre-flop decisions, such as calling a 3-bet or 4-bet all-in. When determining if we can call a pre-flop all-in jam, we take a deep dive look at our opponent’s estimated all-in range to determine how many combinations of starting hands we beat compared to how many we lose against. We then evaluate this compared to our pot odds to determine if calling is +EV or –EV We raise to 3bb in MP with Q♣ Q♦, CO 3-bets to 10bb, we 4-bet to 24bb and CO 5-bet jams all-in for his remaining 94bb stack. CO is a bad aggressive opponent that has been playing somewhat wildly all session. What is CO’s Estimated Jamming Range? Given that CO is a bad aggressive opponent, we know he will be 5-bet jamming a fairly wide range. We estimate that range to include all pocket pairs, KQ+, AJ+, some suited connectors, and some suited Ace bluffs, giving him the following estimated 5-bet jamming range: 22+, AJs+, A5s-A2s, KQs, QJs, JTs, T9s, 98s, 87s, AJo+, and KQo. What is Our Equity versus Villain’s Range? According to Equilab, we are a 69.55% equity favorite versus villain’s range Without doing any pot odds and equity analysis, we already know that calling villain’s pre-flop all-in is a profitable play. We can use Combinatorics to break villain’s range down into combinations to see how much of his actual range we’re ahead of pre-flop <Combinations we beat and which we don't> As you can see from the analysis above, we’re currently ahead of 93% of villain’s pre-flop jamming range. Specifically, we’re ahead of 160 combinations and behind 12 combinations of AA and KK. We’re also a 50/50 equity split with pocket Queens, which is not included in the table. If we’re currently ahead of 93% of villain’s all-in jamming range, then why does Equilab estimate that we have 69.55% equity versus this range? Because our opponent’s hands have post-flop equity against us. Remember that equity changes from pre-flop to post-flop, so while we’re winning now, that doesn’t mean we’ll have the best hand at show down. Equilab estimates we’ll have the best hand at show down 69.55% of the time. In a 30nl game, a good aggressive regular raises to $0.90 pre-flop and hero calls from the SB with A♥ K♠. The flop comes J♠ A♠ K♥, giving hero top two pair on a really wet board texture. Hero checks, villain bets $0.90 into a $2.10 pot and hero calls. The turn card is the 7♦. Hero checks again, villain bets $2.40 into a $3.90 pot and hero calls. The river is the T♠ making the final board J♠ A♠ K♥ 7♦ T♠. Hero checks again, villain bets $7.80 into an $8.70 pot and the action is on hero. Should hero call? What are Hero’s Pot Odds? Hero has to call $7.80 to win the $16.50 pot, giving him 32% pot odds, meaning hero must have 32% equity or better to call here Flush Analysis With the K♠ in his hand, hero blocks K♠ Q♠ Furthermore, villain cannot have a good Q♠ hand, because all other spade Broadway cards are on the board. Accordingly, this rules out all high Broadway combos of flush draws via blockers and card removal. Knowing this, the only likely flush draws villain can have made are suited connectors, such as 9♠ 8♠, 8♠ 7♠, and possibly 6♠ 7♠. This tells us that it’s highly unlikely that villain made a flush on the river. Straights and Worse Made Hands Since villain can only realistically have 3 combinations of made flush draws on the river, it’s much more likely that he has either the nut straight or a set. Given his river bet sizing on such a scary board texture, we should weigh his hand towards Qx (AQ, KQ, and QJ) hands for the nut straight. With a flush and 4-straight on the board, we would expect him to check back sets and two pairs, due to their showdown value and the unlikelihood of getting paid off on such a bad river card for those hands. Assuming that villain would only value bet the nut straight on the river, we can assign him the following combinations of made straights with AQ, KQ and QJ: AQ = 2 Aces x 4 Queens = 8 Combos KQ = 2 Kings x 4 Queens = 8 Combos QJ = 4 Queens x 3 Jacks = 12 Combos Combos Analysis We estimate that villain has 28 combinations of made nut straights and only 3 combinations of made flushes. Call, Raise or Fold? Calling is out of the question. Hero only beats a pure bluff, which doesn’t occur that often at the micro stakes. However, since we weigh villain’s range strongly to made straights with very little combinations of made flushes, this is a potential opportune spot to check/raise jam all-in as a bluff if we think villain will lay down the straight to a flush. Most of the time, we should just be folding here, but against certain opponents, this is a potential bluff spot, especially if stack sizes are deep enough When we balance our ranges, we compose them with equal amounts of value and bluff hands. We can use Combinatorics to balance our value and bluff hands into polarized ranges, such as polarized 3-bet and 4-bet ranges To show you how this works, we’re going to develop a simple 1-to-1 value-to-bluff polarized 3-betting range. To do this, we first need to determine what our 3-bet value range will be composed of, which for this example will be QQ+ and AKs: AA – 6 Combinations KK – 6 Combinations QQ – 6 Combinations AKs – 4 Combinations This gives us a total of 22 combinations of value hands. Next, we add close to 22 combinations of 3-bet bluffing hands. I personally like to use small pocket pairs, suited connectors and small suited Aces, so we’ll add some until we get close to 22 combinations of hands: 22 – 6 Combinations A2s – 4 Combinations A3s – 4 Combinations 87s – 4 Combinations 98s – 4 Combinations We now have a total of 22 combinations of 3-bet bluffing hands, giving us a balanced polarized 3-betting range that looks like the following