# 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