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 longterm 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 preflop raises, isolation raises, squeeze plays, 3bets, 4bets, 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 allin preflop. 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 shortstacked and others deepstacked, 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.

StacktoPot 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 “risktoreward” 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 randomsized 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 riskaverse, and only play the very bestofthebest starting hands preflop. Additionally, they will usually only get involved in big pots postflop with a very strong hand. Most NITs play a very tight and aggressive style of poker, and will play fitorfold postflop. This means that they will only continue with a hand postflop 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 LAGstyle 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 riskadverse, 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 reraise 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 valuebet 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 preflop to try to see flops for as cheap as possible. However, when facing preflop aggression, a loose passive opponent will usually fold. This type of opponent plays in a fitorfold manner postflop, meaning they will fold if they miss out and will often never bluff. A loose passive opponent will only bet or raise preflop and postflop with a strong hand or very strong draw. When you play against a loose passive opponent, you will see him limping in preflop a majority of the time. A passive opponent will only raise preflop 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 preflop, even to raises and reraises with a wide range of hands. Postflop, they will float continuation bets with draws and acehigh 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 buyins, 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 valuebetting. 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 7.5to1 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
 Given 2:1 → m:n, where m = 2 & n = 1
<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 longrun based upon how often we should win.

Let’s use a simple coinflip 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 longrun. 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 coinflipping 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 longterm expectation.

What in the world does that mean?

It means that mathematical variance can cause significant, unexpected results in the shortterm, 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 shortterm 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 preflop 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.
 Preflop allin situations are a common occurrence in poker, so we’ll use a fairly common scenario of QQ versus AK allin preflop. 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 allin 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 allin 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.