Maths
The maths of poker, in the order you need it
Solved Hand6 min read
Poker maths is a short list of sums, and they come in a fixed order. Each one answers the question the last one left open: what a call costs, how often you win, what the call is worth, what you might win later, and when a bet wins without a showdown. This post works through them with the formulas the trainer uses to grade your play.
01What a call costs: pot odds
There is 100 in the pot and your opponent bets 50. To stay in, you put in 50 more. If you win, you collect a pot of 200, and 50 of that is your own call coming back. What you stand to win is the other 150: the original 100 and your opponent's 50. You risk 50 to win 150, and that ratio, 3 to 1, is the .
Pot odds turn into a percentage you can compare with your chances. The call breaks even when you win as often as the price asks, which is your call divided by the pot after you call:
Pot after your call = 100 + 50 + 50 = 200
Equity needed = 50 ÷ 200
= 25%
That 25% is the . Win more often than that and the call makes money over time. Win less often and it loses. Try other numbers here:
- 3.0 : 1
- 25.0%
You call 50.0 to win 150, so the call breaks even at 25.0%.
Bigger bets ask for more. For the same 100 pot:
| Their bet | You call | Equity needed |
|---|---|---|
| A third of the pot | 33 | 20% |
| Half the pot | 50 | 25% |
| Three quarters | 75 | 30% |
| The pot | 100 | 33% |
| Twice the pot | 200 | 40% |
A pot-sized bet asks for a third of the time. A bet of twice the pot asks for 40%, still well short of half, which is why calling a big bet with a hand that wins only some of the time can be right.
02How often you win: equity and outs
Your is the share of the pot you would win, on average, if all the cards were dealt out now with no more betting. Against a made hand, a drawing hand's equity comes from its : the unseen cards that turn it into the winner.
Hold two hearts with two more on the flop and any of the other 9 hearts makes your flush. You can see five cards, so 47 are unseen. The chance that at least one of the next two cards is a heart is:
Chance both cards miss = (38 ÷ 47) × (37 ÷ 46)
Chance at least one hits = 1 − 0.650
= 35.0%
The quick estimate is the rule of 2 and 4: multiply your outs by 4 with two cards to come, or by 2 with one. Nine outs gives an estimated 36% chance of hitting by the river, against the exact 35.0%. The estimate drifts further from the exact figure as your outs go up, because it counts the times you hit on both streets as two hits:
| Draw | Outs | Rule of 4 | Exact, two cards | Exact, one card |
|---|---|---|---|---|
| Gutshot straight draw | 4 | 16% | 16.5% | 8.7% |
| Open-ended straight draw | 8 | 32% | 31.5% | 17.4% |
| Flush draw | 9 | 36% | 35.0% | 19.6% |
| Flush draw and a gutshot | 12 | 48% | 45.0% | 26.1% |
| Flush draw and an open-ended straight draw | 15 | 60% | 54.1% | 32.6% |
Outs assume the draw wins when it hits and loses when it misses. A flush can still lose to a bigger flush, and a pair can still win unimproved, so treat the count as a close estimate, not the exact figure.
03What a call is worth: expected value
Pot odds and equity together give the call's : what it wins or loses on average if you could make the same call many times.
of a call = equity × pot after your call − your call
With 30% equity: 0.3 × 200 − 50 = +10.0
With 20% equity: 0.2 × 200 − 50 = −10.0
Folding is worth zero from here: the chips you put in earlier are gone either way. So a call is right when its EV is above zero, which happens exactly when your equity beats the 25% the price asks for.
That is the idea behind the trainer's free lesson, the call that loses and still pays, which plays it out over hundreds of hands.
04Money still to come: implied odds
Pot odds price the call against the chips in the pot now. When you are drawing, you may also win more on later streets if you hit and your opponent pays you off. Those later chips are your .
Take the same 50 bet into 100, but on the turn with a flush draw. One card is left, so you hit 9 times in 46, or 19.6%, short of the 25% you need. On the pot alone the call loses 10.9 on average. Implied odds say how much more you need to win, on the hands where the flush comes, for the call to break even:
Extra winnings needed when you hit = your call ÷ chance of hitting − pot after your call
= 50 ÷ 0.196 − 200
= about 56 more chips, won on the river when you hit
That is the extra you need from your opponent each time you complete the flush, not on average across every river. Whether you can win that much depends on the stacks behind and how willing your opponent is to pay you off. Deep stacks and a draw that is hard to see make it likely. A short stack, or a third heart that everyone notices, makes it unlikely, and the call goes back to losing.
05Winning without a showdown: bluffs and fold equity
So far you have been the caller. When you bet, you can also win when your opponent folds, and the part of a bet's value that comes from folds is its .
A pure bluff, a hand that never wins when called, risks the bet to win the pot. It breaks even when the opponent folds often enough:
Folds needed = your bet ÷ (pot + your bet)
= 50 ÷ (100 + 50)
= 33.3%
That 33.3% is the . From the other chair, the caller's is the rest, 66.7%: against a 50 bet into 100, the caller has to continue with at least two thirds of their range, or the bettor profits by betting any two cards. It is a theoretical baseline: against a player who rarely bluffs, folding more is still right.
A draw changes the sum, because a bet with outs can still win when it is called. That bet is a . In a simple version of the turn example, where the flush draw wins whenever it hits and nobody bets on the river, betting 50 into 100 breaks even if your opponent folds about 10% of the time. A pure bluff, which never wins when called, needs 33.3%.
06Putting the sums in order
At the table the questions come in the same order every time:
- What is the price? Your call divided by the pot after you call.
- How often do I win? Your equity, from outs on a draw or from your read of their range.
- Which is bigger? If your equity beats the price, the call has positive EV.
- What changes later? Implied odds for a call, fold equity for a bet.
Each is a line or two of arithmetic. The hard part is doing them every time, quickly, and that is what the trainer's drills practice.
Try itPoker math, every formulaPot odds, EV, outs, implied odds, ICM and more, each with what its symbols mean and a worked example.Try itThe call that loses and still paysA two-minute interactive lesson: decide whether to call, then watch the price play out over hundreds of hands.Sources
- Poker math: pot odds, equity, EV and ICM formulas, Solved Hand
- Pot odds, Wikipedia
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