Poker math: pot odds, equity, EV and ICM formulas

Odds and value

Every decision is priced in expected value. These are the sums you do most often.

The average result of a decision if you could make it many times over: each outcome's result weighted by how often it happens.

\text{EV} = \sum_i p_i \, x_i

$\sum_i$: add up over every outcome $i$
$p_i$: the chance of outcome $i$ ; the chances add up to 1
$x_i$: what you win or lose in outcome $i$ , in big blinds

Worked example

You bet 10bb into 15bb. Your opponent folds 40% of the time and you take the pot; otherwise they call and your bluff loses the bet.

\text{EV} = 0.4 \times 15 + 0.6 \times (-10) = 6 - 6 = 0\text{bb}

Why it matters

Every other formula on this page is expected value in disguise. A decision is right when its expected value beats the alternatives, whatever happens this time.

The price the pot offers you to call, and the share of the time you must win for the call to break even.

\text{pot odds} = P : C

e_{\text{req}} = \dfrac{C}{P + C}

$P$: the pot, including your opponent's bet
$C$: the amount you have to call
$e_{\text{req}}$: required equity: the share of the time the call must win

Worked example

The pot is 100bb and your opponent bets 50bb.

P = 100 + 50 = 150, \quad C = 50

\text{pot odds} = 150 : 50 = 3 : 1

e_{\text{req}} = \dfrac{50}{150 + 50} = 25\%

Why it matters

Compare it with your equity: call when your equity is above the requirement, fold when it is below. Bigger bets demand more equity.

How much a call wins or loses on average, not just whether it is right. Setting it to zero gives back the required equity.

\text{EV}_{\text{call}} = e \, (P + C) - C

e \, (P + C) - C = 0 \;\Rightarrow\; e = \dfrac{C}{P + C}

$e$: your equity: your share of the pot at showdown
$P, C$: the pot including the bet, and the amount to call

Worked example

Facing the same 50bb bet into 100bb, you have 30% equity.

\text{EV}_{\text{call}} = 0.3 \times (150 + 50) - 50 = 60 - 50 = +10\text{bb}

Why it matters

It tells you how much a mistake costs. That is the number the grades in this app are built on.

The cards that complete your draw, and the chance one of them arrives. The rule of 2 and 4 is the quick estimate; the exact sum counts the unseen cards.

P_{\text{flop to river}} = 1 - \dfrac{(47 - o)(46 - o)}{47 \times 46} \approx 4o\%

P_{\text{turn to river}} = \dfrac{o}{46} \approx 2o\%

$o$: the number of outs
$47, 46$: the unseen cards on the flop and on the turn: 52 less your 2 and the board

Worked example

A flush draw has 9 outs.

\text{Flop: } 1 - \dfrac{38 \times 37}{2162} = 35\% \quad (\text{rule: } 36\%)

\text{Turn: } \dfrac{9}{46} = 19.6\% \quad (\text{rule: } 18\%)

\text{15 outs on the flop: } 54.1\% \quad (\text{rule: } 60\%)

Why it matters

The rule is close enough at the table for up to about 9 outs. With more it overstates: 15 outs is 54.1%, not 60%, because the formula counts hitting twice as once.

Money you expect to win on later streets when your draw hits. It lets you call with less equity than the pot odds alone demand.

e \, (P + C + X) \ge C \;\Rightarrow\; X \ge \dfrac{C}{e} - (P + C)

$X$: the extra you must win, on average, the times you hit
$e$: the chance your draw hits
$P, C$: the pot including the bet, and the amount to call

Worked example

On the turn with a flush draw (9 outs), you face a 25bb bet into 50bb. You need 25% and hit 19.6% of the time.

X \ge \dfrac{25}{0.196} - (75 + 25) = 127.8 - 100 = 27.8\text{bb}

Why it matters

If you can expect to win another 28bb or more when the flush comes in, the call pays. Implied odds need deep stacks and a hidden draw; they vanish when the opponent is short or the draw is obvious.

Betting

The same indifference sums, seen from the bettor's side and the caller's.

How often a bet must fold out the opponent to profit. A semi-bluff also wins some of the times it is called, so it needs fewer folds.

f_{\min} = \dfrac{B}{P + B}

\text{EV}_{\text{semi-bluff}} = f\,P + (1 - f)\big(e\,(P + 2B) - B\big)

$B$: your bet
$P$: the pot before you bet
$f$: how often the opponent folds
$e$: your equity when called
$f_{\min}$: the fewest folds that let a pure bluff break even

Worked example

You bet 75bb into 100bb.

\text{Pure bluff: } f_{\min} = \dfrac{75}{100 + 75} = 42.9\%

\text{Half-pot bluff: } f_{\min} = \dfrac{50}{100 + 50} = 33.3\%

\text{With 20\% equity when called: } e\,(P + 2B) - B = 0.2 \times 250 - 75 = -25

f\,100 + (1 - f)(-25) = 0 \;\Rightarrow\; f = 20\%

Why it matters

Equity turns a bet that needs 42.9% folds into one that needs only 20%. That is why draws make the best bluffs.

The share of your range you must continue with so that the opponent cannot profit by bluffing with any two cards.

\text{MDF} = \dfrac{P}{P + B} = 1 - f_{\min}

$P$: the pot before the bet
$B$: the bet you face
$f_{\min}$: the folds a pure bluff of that size needs to break even

Worked example

You face a 75bb bet into 100bb.

\text{MDF} = \dfrac{100}{100 + 75} = 57.1\%

Why it matters

Fold more than 42.9% of your range and every bluff your opponent makes shows a profit. Against players who rarely bluff, folding more is still right; the formula is a defense against the ones who do.

On the river, the share of bluffs in a betting range that leaves the caller indifferent: their required equity against it is exactly met.

\text{bluff share} = \dfrac{B}{P + 2B}

\text{bluffs} : \text{value} = B : (P + B)

$B$: your river bet
$P$: the pot before you bet

Worked example

Two river sizes into a 100bb pot.

\text{Pot-sized bet: } \dfrac{100}{100 + 200} = 33.3\%, \quad 1 : 2

\text{75bb bet: } \dfrac{75}{100 + 150} = 30\%, \quad 3 : 7

Why it matters

Bigger bets can carry more bluffs, because the caller needs more equity. Too many bluffs and you get called; too few and you get folded to.

How many pots are left behind on the flop. It tells you how committed a hand is before you bet a chip.

\text{SPR} = \dfrac{S}{P}

$S$: the effective stack behind, the smaller of the two stacks
$P$: the pot on the flop

Worked example

Button against big blind, 100bb deep.

\text{Single-raised pot: } \dfrac{97.5}{5.5} = 17.7

\text{3-bet pot: } \dfrac{89}{22.5} = 4

Why it matters

At a ratio near 4, top pair is usually happy to get all the chips in. Near 18 it is a hand to control the pot with, not to stack off.

The single pot fraction that, bet and called on every street left, puts all the chips in on the river.

P\,(1 + 2f)^n = P + 2S \;\Rightarrow\; f = \dfrac{(1 + 2S/P)^{1/n} - 1}{2}

$f$: the bet as a fraction of the pot
$n$: the streets left to bet
$S, P$: the effective stack and the pot

Worked example

The pot is 10bb with 50bb behind and 3 streets to go.

f = \dfrac{(1 + 2 \times 50/10)^{1/3} - 1}{2} = \dfrac{2.224 - 1}{2} = 61.2\%\text{ of the pot}

Why it matters

It is the most even way to grow the pot with a strong hand: the opponent faces the same price on every street and never a sudden all-in.

Ranges and combinations

Counting the hands an opponent can hold, and updating the count as they act.

How many ways a hand can be dealt. Ranges are weighed in combinations, not in hand names, because some hands come up far more often than others.

\binom{n}{k} = \dfrac{n!}{k!\,(n - k)!}

\text{pair} = \binom{4}{2} = 6 \qquad \text{unpaired} = 4 \times 4 = 16

$n$: the cards available
$k$: the cards drawn
$n!$: $n$ factorial: $n \times (n - 1) \times \dots \times 1$

Worked example

Ace-king against a pocket pair.

\text{All starting hands: } \binom{52}{2} = 1{,}326

\text{AK: } 16 = 4\text{ suited} + 12\text{ offsuit}, \qquad \text{KK: } 6

Why it matters

Ace-king is dealt more than twice as often as kings. A range of value hands and bluffs is only balanced once you count it in combinations.

Every card you can see is a card your opponent cannot hold. Removing it shrinks the combinations of every hand that needs it.

\text{pair} = \binom{r}{2} \qquad \text{unpaired} = r_1 \times r_2

$r$: how many cards of that rank are still unseen

Worked example

You hold the ace of spades.

\text{AA: } \binom{3}{2} = 3 \text{ (was 6)}

\text{AK: } 3 \times 4 = 12 \text{ (was 16)}

Why it matters

Holding an ace halves the aces your opponent can have. That is why an ace blocker makes a good bluff: it removes the hands most likely to call.

Bayes' rule: after an action, each hand in the range is reweighted by how often it would take that action.

P(H \mid A) = \dfrac{P(A \mid H)\,P(H)}{\sum_j P(A \mid H_j)\,P(H_j)}

$H$: a group of hands, such as the bluffs
$A$: the action you saw, such as a river bet
$P(H \mid A)$: the share of the range that is $H$ once you have seen $A$

Worked example

Your opponent reaches the river with 30 value combos and 10 missed draws. They bet every value hand and half their draws, and bet the pot. Your hand beats only the bluffs.

P(\text{bluff} \mid \text{bet}) = \dfrac{0.5 \times 10}{1 \times 30 + 0.5 \times 10} = \dfrac{5}{35} = 14.3\%

\text{Pot-sized bet: } e_{\text{req}} = 33.3\% > 14.3\% \;\Rightarrow\; \text{fold}

Why it matters

Before the bet 25% of the range were bluffs; after it only 14.3% are, and 85.7% are value. Reading hands is this sum done by feel on every street.

Tournaments

Short stacks and prize pools, where chips stop being worth the same.

With a short stack the small blind either moves all in or folds. Shoving wins the blinds whenever the big blind folds, and plays a showdown when it calls.

\text{EV}_{\text{push}} = (1 - c)(1 + a) + c\,S\,(2e - 1)

\text{EV}_{\text{fold}} = -(0.5 + a)

$S$: the effective stack in big blinds, ante included
$a$: the ante
$c$: how often the big blind calls
$e$: your equity when called

Worked example

Heads-up in the small blind with 10bb and no ante. The big blind calls 37.4% of the time, and you have 40% equity when called. Results are measured from before the blinds go in.

\text{EV}_{\text{push}} = 0.626 \times 1 + 0.374 \times 10 \times (2 \times 0.4 - 1) = -0.12\text{bb}

\text{EV}_{\text{fold}} = -0.5\text{bb} \;\Rightarrow\; \text{push}

Why it matters

The shove loses money on average and is still right, because folding loses more. The push/fold charts solve this for every hand at once, each side indifferent at the edge of its range.

Turns chip stacks into shares of the prize pool. A player finishes first with the chance of their share of the chips, and each later place is worked out the same way among the players left.

P_i(1) = \dfrac{s_i}{T}

P_i(2) = \sum_{j \ne i} \dfrac{s_j}{T} \cdot \dfrac{s_i}{T - s_j}

\text{prize equity}_i = \sum_k \text{prize}_k \, P_i(k)

$s_i$: player $i$ 's stack
$T$: all the chips in play
$\text{prize}_k$: the payout for finishing in place $k$
$P_i(k)$: the chance player $i$ finishes in place $k$

Worked example

Three players have 50, 30, 20 of 100 chips. The prizes are 50, 30, 20.

P_{\text{big}}(2) = \dfrac{30}{100} \cdot \dfrac{50}{70} + \dfrac{20}{100} \cdot \dfrac{50}{80} = 33.9\%

\text{prize equity} = 38.4, \;32.8, \;28.9

\text{chip share} = 50, \;30, \;20

Why it matters

The chip leader holds half the chips but only 38.4% of the prizes. Chips you lose cost more than chips you win are worth, so calls near a pay jump need more equity than chip counting says.

Grades and results

How this app scores a decision, and how long it takes for results to mean anything.

The expected value you gave up against the best action, as a share of the pot. Every graded decision with solver or push/fold values is scored this way.

\ell = \dfrac{\text{EV}_{\text{best}} - \text{EV}_{\text{chosen}}}{P} \times 100\%

\text{grade} = \begin{cases} \text{Best} & \ell \le 0.25\% \\ \text{Correct} & \ell \le 2\% \\ \text{Inaccuracy} & \ell \le 5\% \\ \text{Wrong} & \ell \le 15\% \\ \text{Blunder} & \text{above} \end{cases}

$\ell$: the loss as a share of the pot
$P$: the pot at the decision

Worked example

In a 10bb pot raising is worth +2bb and calling +1.6bb. You call.

\ell = \dfrac{2 - 1.6}{10} \times 100\% = 4\% \;\Rightarrow\; \text{Inaccuracy}

Why it matters

Measuring against the pot keeps a small slip in a big pot and a big slip in a small pot on one scale. Per 100 decisions, it is the closest thing to a winrate a drill can measure.

A winrate measured over a sample is an estimate. Its margin of error shrinks with the square root of the number of hands.

w \pm z \, \dfrac{\sigma}{\sqrt{N / 100}}

N_{\text{needed}} = 100 \left(\dfrac{z\,\sigma}{w}\right)^2

$w$: the measured winrate, in big blinds per 100 hands
$\sigma$: the standard deviation per 100 hands, often 70 to 100 in no-limit
$N$: hands played
$z$: 1.96 for 95% confidence

Worked example

You win 5bb per 100 hands over 20,000 hands, with a standard deviation of 80.

5 \pm 1.96 \times \dfrac{80}{\sqrt{200}} = 5 \pm 11.1 \;\Rightarrow\; [-6.1,\ 16.1]

N_{\text{needed}} = 100 \left(\dfrac{1.96 \times 80}{5}\right)^2 \approx 98{,}345\text{ hands}

Why it matters

20,000 hands cannot tell a winning player from a losing one. Judge your play by the decisions graded here, not by short-term results.

The chance a bankroll eventually hits zero, for a player who keeps playing with the same winrate and swings.

\text{RoR} = e^{-2 w B / \sigma^2}

B = \dfrac{-\sigma^2 \ln(\text{RoR})}{2w}

$w, \sigma$: winrate and standard deviation, both per 100 hands
$B$: the bankroll, in big blinds
$e, \ln$: Euler's number, about 2.718, and the natural logarithm; here $e$ is not equity

Worked example

The same player, 5bb per 100 hands with a standard deviation of 80, has 2,000bb: 20 buy-ins of 100bb.

\text{RoR} = e^{-2 \times 5 \times 2{,}000 / 80^2} = e^{-3.125} = 4.4\%

\text{For } 1\%: \; B = \dfrac{80^2 \times \ln 100}{2 \times 5} = 2{,}947\text{bb} \approx 29.5\text{ buy-ins}

Why it matters

A winning player can still go broke. Halving the risk takes a bigger roll, and the formula only holds while the winrate does; move down in stakes before the roll runs thin.

Odds and value

Expected value

Worked example

Why it matters

Pot odds and required equity

Worked example

Why it matters

The value of a call

Worked example

Why it matters

Outs and the rule of 2 and 4

Worked example

Why it matters

Implied odds

Worked example

Why it matters

Betting

Bluffs and fold equity

Worked example

Why it matters

Minimum defense frequency

Worked example

Why it matters

Bluff-to-value ratio

Worked example

Why it matters

Stack-to-pot ratio

Worked example

Why it matters

Geometric bet sizing

Worked example

Why it matters

Ranges and combinations

Counting combinations

Worked example

Why it matters

Card removal and blockers

Worked example

Why it matters

Updating a range

Worked example

Why it matters

Tournaments

Push or fold

Worked example

Why it matters

The independent chip model

Worked example

Why it matters

Grades and results

EV loss and grades

Worked example

Why it matters

How sure is a winrate?

Worked example

Why it matters

Risk of ruin

Worked example

Why it matters