Intro to the Mathematics of Poker
90% of the book in under 15 minutes.
The goal of poker should be to maximize Expected Value (EV).
EV is calculated by taking into account all possible outcomes of a decision and weighting the value of each outcome by its probability. To calculate EV, it’s necessary to understand how to determine probability.
The probability of an event is the number of ways the event can occur divided by the total number of possible outcomes.
For example, the probability of drawing an ace from a standard deck of 52 cards is 4/52, or 1/13, because there are four aces and 52 total cards.
If multiple events are mutually exclusive (meaning they cannot both happen), then the probability of either event occurring is the sum of their individual probabilities.
For example, the probability of drawing an ace or a king from a deck of cards is 8/52, or 2/13, because there are four aces, four kings, and 52 total cards.
If the events are dependent (meaning the probability of one event occurring impacts the probability of the other event occurring), then the probability of both events occurring is the product of their individual probabilities.
For example, the probability of drawing two aces in a row from a deck of cards (without replacing the first card) is (4/52) * (3/51), or 1/221.
A probability distribution is a function that describes the probability of each possible outcome of a random variable.
For example, the probability distribution for flipping a fair coin twice is:
Heads, Heads: 1/4
Heads, Tails: 1/4
Tails, Heads: 1/4
Tails, Tails: 1/4
The EV of a probability distribution is the sum of the products of the values of each outcome and their respective probabilities.
Formula: EV = (%W * $W) + (%L * $R), where:
%W = Probability of winning (equity)
$W = Amount you can win
%L = Probability of losing
$R = Amount you risk
For example, the EV of a $1 bet on a coin flip (assuming you win $1 if you guess correctly and lose $1 if you guess incorrectly) is:
($1) * (1/2) + (-$1) * (1/2) = $0
This means that, on average, you will neither win nor lose money by betting on a coin flip.
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In poker, equity (Eq) represents a player's share of the pot based on their current chance of winning or splitting the pot at a particular point in the hand. It essentially answers the question, "If the hand were to end right now and all cards were revealed, how often would I win the pot on average?".
Equity is a crucial concept for making sound decisions in poker because it helps players understand the value of their hand relative to their opponents' hands or range of possible hands.
Here are some important points about poker equity:
Equity is not fixed: A hand's equity changes dynamically throughout the hand as the board develops and more cards are revealed.
Equity depends on the opponent's hand: A hand's strength cannot be evaluated in isolation; it's always relative to what the opponent is holding.
Hand vs Hand Equity: This compares the equity of one specific hand against another specific hand. For example, comparing the equity of pocket aces (AA) versus pocket kings (KK). While useful for understanding basic hand matchups, it's limited because it doesn't account for the range of hands an opponent might be playing.
Hand vs Range Equity: This compares the equity of a specific hand against a range of hands that the opponent might be holding, based on their actions and tendencies. For example, calculating the equity of your hand against a tight player who has raised pre-flop.
Range vs Range Equity: This compares the equity of your range of possible hands against the opponent's range of possible hands. This is particularly useful in post-flop scenarios where the community cards have influenced the strength of both players' ranges.
In poker, outs are the cards that a player needs to improve their hand to a winning hand. Outs are directly related to equity, which is a player's share of the pot based on their chance of winning. The more outs a player has, the higher their equity will be.
Calculating Equity from Outs:
There's a general rule of thumb for estimating equity based on the number of outs:
On the flop (with two cards to come): Multiply the number of outs by 4. This gives a rough estimate of the percentage chance of hitting one of your outs on either the turn or the river.
On the turn (with one card to come): Multiply the number of outs by 2. This estimates the percentage chance of hitting one of your outs on the river.
For example, if you have a flush draw on the flop (9 outs), your approximate equity is 9 x 4 = 36%.
It's important to consider "dead outs" when calculating equity. Dead outs are cards that would complete your desired hand but would also improve your opponent's hand to a better hand. This reduces your effective outs and lowers your equity. For instance, if you have an open-ended straight draw but your opponent has a flush draw, two of your outs will also give your opponent a flush, leaving you with fewer live outs.
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Pot odds refer to the ratio of the current size of the pot to the cost of calling the bet. For instance, if the pot is $100 and an opponent bets $25, the pot odds are 4:1. This means that a player needs to win at least 1 out of every 5 times to break even on a call. To determine whether a call is profitable using pot odds, a player needs to compare the pot odds to their chance of winning the hand if they call. If their chance of winning is greater than the pot odds, calling is profitable; otherwise, folding is generally the better option.
Implied odds take into account the potential for winning additional money on future betting rounds if the draw is completed. This concept recognizes that in real poker games, players don't always have perfect information about their opponents' hands, and made hands may sometimes call bets even after a draw is completed. This can lead to situations where a draw can profitably call a bet even if the immediate pot odds aren’t favorable, because they expect to win more money from their opponent on future streets if their draw comes in.
Here’s a key difference between pot odds and implied odds:
Pot odds are based solely on the current size of the pot and the cost of calling.
Implied odds take into account potential future betting and the possibility of winning more money after the draw is complete.
Example: Imagine a situation where a player has a flush draw on the flop. The immediate pot odds might not justify a call, but if the player believes their opponent will call a significant bet on the turn and river if the flush comes in, the implied odds could make the call profitable. Essentially, the player is "investing" a smaller amount now with the expectation of winning a much larger amount later.
A pure bluff is a bluff that has no chance of winning at showdown and no chance to improve.
A semi-bluff is a bet with a hand that may be currently the best but has outs to substantially improve on later streets.
A value bet is a bet that has +EV when called by villain’s maximally exploitative strategy.
A strategy is a complete plan that dictates a player's actions at every possible decision point throughout the game.
Optimal strategies are strategies that maximize expectation against a "nemesis," a hypothetical opponent who always knows your strategy and plays to maximally exploit it. Optimal strategies in poker do not include "loss leaders," or plays that sacrifice immediate expectation for deceptive purposes. Instead, if a hand is played in different ways, each way of playing that hand must have the same expectation against the opponent's optimal strategy. Otherwise, the player could unilaterally improve by shifting to the higher-expectation option.
Optimal strategies for zero-sum two-player games have these properties:
They always exist as long as mixed strategies are allowed (meaning each player can use strategies such as, "do X 60% of the time and Y 40%").
They do not contain strictly dominated alternatives. This means that there is no other option that always performs better against all opponent strategies.
In a mixed strategy, the expectation of each strategic alternative must be equal against the opponent's optimal strategy.
Optimal strategy pairs in a zero-sum two-player game have the following qualities:
Neither player can improve their expectation by unilaterally changing their strategy.
They consist of two strategies that maximally exploit each other.
This definition of "optimal" is narrow and specific to game theory strategy.
In optimal poker play, bluffing frequency is inversely related to pot size: as the pot grows larger, the optimal bluffing frequency decreases.
This might seem counterintuitive, given that there is more to gain from a successful bluff into a larger pot. However,optimal bluffing is not about making bluffing profitable on its own. Rather, optimal bluffing is part of a larger strategy of balancing bluffs and value bets in a way that ensures a positive expected value (EV) regardless of the opponent's response.
Here’s how this works:
Opponents who fold too often will lose value to bluffs.
Opponents who call too often will lose value to value bets.
By balancing bluffs and value bets correctly, the optimal player can exploit either tendency in their opponent.
Alpha (α), represents the ratio of bluffs to value bets needed to keep the opponent indifferent between calling and folding.
In games with variable bet sizes, α = s : (1 + s), where s is the bet size as a fraction of the pot.
For a bet size of 50%, α = 1 : 3, allowing up 1 bluff for every 3 value bets.
For a bet size of 100%, α = 1 : 2, allowing 1 bluff for every 2 value bets.
This formula demonstrates that:
As pot size increases, s decreases.
As s decreases, α also decreases.
When used as a fraction (α = s / ( 1 + s)), α represents how often an opponent should fold to s bet size to prevent a bluff from being automatically profitable.
This means players should bluff less often when betting into larger pots. While there’s more to gain from a successful bluff into a large pot, the risk of losing a large bet when called outweighs the potential gains from bluffing more often.
Optimal play is about finding the right balance between bluffing and value betting to maximize EV over the long run. This involves understanding how pot size affects bluffing frequency and adjusting your strategy accordingly.
In poker, balance refers to the degree to which a strategy is unexploitable by opponents. Strategies that can be heavily exploited are considered unbalanced, while those that are difficult or impossible to exploit are considered balanced. Strategies that are balanced are not often the most profitable strategy, but they are often very strong.
A balanced strategy aims to minimize weaknesses that opponents can easily identify and take advantage of. This means avoiding predictable patterns, like always betting with strong hands and checking with weak ones, which would allow opponents to easily adjust their play to exploit these tendencies.
A perfectly balanced strategy makes the opponent indifferent to the various options they have at each decision point. The goal is to create a situation where no matter what action the opponent chooses, they cannot gain a significant EV advantage.
For example, if Player A bets the same amount on the river with both their strong value hands and their bluffs in the correct ratio, Player B will be indifferent to calling or folding, as their EV will be roughly the same regardless of their choice. This is because they will win often enough against bluffs to offset their losses against value bets, and vice versa. This principle is also related to the concept of alpha (α), the ratio of bluffs to value bets needed to keep the opponent indifferent.
A balanced strategy often provides flexibility to the player using it. This means they can make small mistakes near threshold points without significantly impacting their EV. This is because their balanced ranges make the opponent relatively indifferent to different actions.
Conversely, an opponent trying to exploit a balanced strategy often faces a brittle situation. Small errors in their reads or adjustments can lead to significant EV losses.
Exploitative strategy in poker centers around deviating from a balanced, theoretically optimal approach (like Game Theory Optimal, or GTO) to capitalize on an opponent's identifiable weaknesses or tendencies. Rather than aiming for an unexploitable strategy, it seeks to maximize expected value (EV) by directly attacking vulnerabilities in how an opponent plays.
Maximally exploitative strategies often require extreme swings away from optimal, and adjustments will be noticed by sharper opponents, who will then counter-exploit. Extreme strategies are often extremely profitable until they are extremely unprofitable.
The heart of exploitative play lies in four principles:
If an opponent bluffs too much, all pure bluff-catchers are +EV.
If an opponent bluffs too little, all pure bluff-catchers are -EV.
If an opponent folds too much, all bluffs are +EV.
If an opponent calls too much, all bluffs are -EV.
Much of exploitative play boils down to asking “Is my opponent doing X too much?” Even if your opponent is over-doing one action by a slight amount, a maximally exploitative strategy will take an extreme swing as a counter-strategy.
Minimum Defense Frequency (MDF) is a key concept that plays a crucial role in understanding exploitative play. MDF represents the minimum frequency at which a player must defend (call or raise) against an opponent's bets to prevent them from profitably bluffing.
MDF is calculated by 1 - α.
For a bet size of 33%, α = 25% and MDF = 75%.
For the bet size of 50%, α = 33% and MDF = 66%.
For a bet size of 100%, α = 50% and MDF = 50%.
This means the smaller the bet relative to the pot, the higher the MDF, and the more often the player must defend.
For example:
Imagine an opponent consistently c-bets the flop with a sizing that requires a 50% MDF, but they are only calling raises with 30% of their hands. This indicates they are significantly over-folding to flop raises. An exploitative strategy would involve raising their c-bets more frequently, even with weaker hands, knowing that the opponent is likely to fold.
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When we can clairvoyantly see both player’s hands, in made hand versus draw situations, the made hand usually bets. The made hands wants to immediately extract value from the draw, as well as follow through with a bet on later streets when the draw frequently misses.
A draw usually calls if it is +EV after calling and subtracting the cost of the call.
$10/$20 No-Limit Hold'em game.
Player A has A♠ Q♦.
Player B has 8♣ 7♣.
The flop is A♣ J♠ 4♣, giving Player A top pair and Player B a flush draw.
The pot is $50. Player A bets $20.
To determine whether Player B should call, we need to consider their equity in the pot if they call:
Potential Winnings: If Player B calls and completes their flush, they will win the pot, which will be $90 ($50 + $20 + $20).
Cost of Calling: Player B needs to call $20 to stay in the hand.
To have positive equity, Player B's chance of winning the pot multiplied by the potential winnings must be greater than the cost of calling. In this case:
Chance of Winning: Player B has 8 outs to complete their flush (any club). With two cards left to come (the turn and river), their chance of winning is roughly 32% (using the rule of 4 and 2).
Expected Value (EV) of Calling:
EV = (Chance of Winning x Potential Winnings) - Cost of Calling
EV = (0.32 x $90) - $20
EV = $28.80 - $20
EV = $8.80
Since the EV of calling is positive ($8.80), Player B has positive equity and should call the bet.
Good draws (those with close to 50% equity) benefit greatly from getting all the money in on the flop. However, if they cannot get all the money in, they prefer to get as little as possible. This is because if a good draw misses on the first card, it can be punished by made hands on later betting rounds.
Some draws (or made hands) have so much equity that they will never be indifferent to a bet size. When choosing which draw to make indifferent, the answer is often the strongest draw that you can.
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The stack-to-pot ratio (SPR) plays a crucial role in determining optimal bet sizing in poker. The larger the SPR, the more money each player can effectively lose later in the hand.
Here's a breakdown of how SPR influences bet sizing:
Low SPR (0-5):
Smaller bet sizes are generally more effective in low SPR situations. This is because the potential gains from bluffing are limited, as the opponent doesn't have much to lose by calling. Additionally, larger bets risk over-committing the player, leaving them with few options on later streets.
Hands that can flop strong hands, like top pairs or overpairs, perform better in low SPR pots. This is due to the decreased fold equity and implied odds in such situations.
Medium SPR (6-11):
A wider range of bet sizes becomes viable as the SPR increases. Players can utilize both smaller bets to induce bluffs and larger bets to extract value from weaker hands. The increased SPR allows for more maneuvering and a greater emphasis on implied odds.
As the SPR moves into the medium range, the value of top pair type hands decreases, while speculative hands with greater implied odds gain value. The emphasis shifts towards suitedness and connectedness rather than high card value.
High SPR (11+):
Larger bet sizes become increasingly optimal as the SPR gets higher. Deep stacks provide more opportunities for value betting and bluffing. Larger bets can effectively deny equity to weaker hands, forcing folds, and extract maximum value from hands that reach showdown.
In high SPR scenarios, the value of hands primarily comes from their "nuttiness," or their potential to make the strongest possible hand. Hands like sets, nut draws, and high flushes become more valuable as they offer the possibility of "coolering" an opponent.
General Principles:
Bet sizing should be proportional to the SPR. Deeper stacks allow for a wider range of bet sizes, including larger bets to extract value and bluff effectively. Shallow stacks limit bet sizing options and typically favor smaller bets.
Larger bet sizes are more effective when a player has a range advantage (a stronger range than their opponent). This allows them to leverage their equity and put more pressure on the weaker range.
SPR also affects the types of hands that are profitable to play. In low SPR scenarios, hands with immediate showdown value are preferred. As SPR increases, hands with greater implied odds and potential to improve gain value.
There is a clear connection between range polarization and bet sizing. Range polarization refers to the difference in strength between the best and worst hands in a player's range. A highly polarized range has very strong hands and very weak hands, but few hands in between. A depolarized range has a more even distribution of hand strengths.
When a player has a highly polarized range, they are incentivized to use larger bet sizes. This is because they can confidently bet large with their strongest hands, knowing they have the best hand often, and they can also bluff more often with their weakest hands, forcing their opponent to fold more often.
Conversely, when a player has a depolarized range, they are more likely to use smaller bet sizes. This is because they have fewer strong hands to value bet with and are more vulnerable to being raised by their opponent if they bluff too much.
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The AKQ Game
This is a super simplified poker game involving only three cards: Ace, King, and Queen. Two players are each dealt one card at random, and one card is removed. There’s one street of betting.
Setup:
Both players ante $1.
One player is dealt A, K, or Q at random. The other player gets one of the remaining two cards.
There’s one betting round. Player 1 can check or bet $1. Player 2 can call or fold.
Key points:
The Ace always beats the King and Queen.
The King beats the Queen.
Queen is the weakest hand.
Now here's the trick: there’s no value in betting with Queen unless you can bluff. But if you never bluff with Queen, then your opponent will never call with King. So to get value from your Ace (when you bet), you must sometimes bluff with Queen. That way, when you do bet, your opponent can't just fold everything but Ace. This is where bluff-to-value ratios come in.
The AKQ game teaches:
Why we bluff weak hands (to make our strong hands get paid).
Why middling hands (like King) are forced to bluff-catch.
How balance forces your opponent into tough spots.
That bluffing is necessary, not optional, even in very simple games.
It’s a bare-bones model of bluffing frequency: if you bluff too often with garbage (the Q), they call more. If you bluff too little, they fold more, and you lose value with your A. The goal is to find the precise balance point that maximizes EV without becoming exploitable.
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The 0-1 Game:
The 0-1 Game is a toy game used to demonstrate optimal play with incomplete information and bluff and value thresholds.
Setup:
$100 in put into the pot.
Player 1 is given a discrete number between 0.99. Player two receives another number.
There’s one betting round. Player 1 can check or bet $100. Player 2 can call or fold. The lowest number wins.
What is the optimal betting strategy for player 1, and what is the optimal calling strategy from player 2?
Answer: Player 1 should bet the top 22% of his range for value (0-.22), and the bottom 11% of his range as a bluff (.88-.99). Player 2 should call with the top 44% of his range (0-.44).
You can simulate this question in PIOsolver by giving each player nine pocket pairs on a board of 22223. The correct attacking strategy is to value bet the top 22% of our range and bluff the bottom 11% of our range.
Opponent’s calling range: the top ~44% of hands.
What if the attacking player only bets top 22% for value and never bluffs?
We should only call with AA.
What if the attacking player bets the top 22% and bottom 14% of hands (slightly overbluffs)?
Player 2 should call with all bluff-catchers that beat Player 1’s bluffs.
What do we learn from the 0-1 game?
--Value and bluffing thresholds are related to pot odds. When we bet 100% on the river, our bluff to value ratio should be 1:2. The larger we bet, the more bluffing combos we can include.
--When we play optimally, our opponent is “indifferent” with some hands between calling and folding, and must correctly mix calling with different bluff-catchers to prevent becoming too predictable, as well as correctly meet the calling frequency.
--When we over-bluff or under-bluff, our opponents should wildly deviate from optimal, not make small adjustments themselves.
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Quote from the Mathematics of Poker:
There are two principles at the core of analyzing multi-street play. The first is that in order for a bluff to be credible, it must come from a distribution of hands that contains strong hands as well. The second principle is also of prime importance and understanding how to apply Game Theory to poker. It is that multi-street games are not single-street games chained together; the solution to the full game is often quite different from the solution to individual streets.
The correct strategy on one street is not isolated from the future; rather, the value of a line is deeply connected to how it interacts with future actions. Solving each street as if it exists in a vacuum leads to significant errors, because decisions on one street set the context for future play. The solver's solution to the full game often deviates from a naïve chaining of optimal one-street solutions. For example, a line that loses a small amount of EV on the flop may enable higher EV realization on later streets through better range protection or increased bluffing opportunities. Similarly, an opponent's turn fold frequency may dictate a mixed strategy on the flop that would look suboptimal if evaluated in isolation.
For every action sequence, there should be corresponding balancing regions. Value, semi-bluffing (when there are cards to come) and bluffing regions should generally be seen together in a strategy.
If a player has a distribution that contains only draws and does not threaten to have a strong hand in a particular situation, it is typically very bad for his expectation.
At every point in your strategy, your range must be composed of made hands and draws so that across all actions and boards, you will have very high equity hands. For example, on T86ss, we must check back some hands that can calldown relentlessly on blanks, as well as some spades, 9x, and 7x so that we can represent flushes and straights on future streets.
Sometimes it may seem that you are being exploited on a particular street while playing near optimally. However, this often occurs because the opponent has sacrificed equity on a previous street in order to obtain the position where he can exploit on this street.
Can it be a mistake to call a 3-bet or 4-bet with KK? Yes, but only if your opponent has built their range in such a way that it makes significant mistakes in other parts of their strategy. If you are getting destroyed in one part of the game tree, your opponent must be leaking in others.
In every action sequence, provided there is a significant amount of action remaining, it is imperative to threaten to have a strong hand so that you cannot be exploited by the opponent wantonly over-betting the pot.
If the stack-to-pot ratio remains high, in every line you must have strong hands so that you can call-down giant bets and raises. For example, if your strategy incorporates both bets and checks on the flop, you must put very strong hands in both your betting and checking lines so that you are protect against flop check-raises (after betting) and turn overbets (after checking).
Similarly, when facing a bet with significant money remaining, you must include nutted hands in both your calling and raising range.