Poker Odds and Equity: A Probability Primer
Poker is not a casino game in the usual sense: you play against other players, not against the house, so skill decides who wins over time. Much of that skill is fast, approximate probability. These are the exact numbers behind the shortcuts.
Starting hands
In Texas Hold'em there are C(52,2) = 1,326 possible two-card starting hands.
| Event | Exact | Approx. |
|---|---|---|
| Dealt any pocket pair | 3/51 | 5.9% (1 in 17) |
| Dealt pocket aces | 6/1,326 | 0.45% (1 in 221) |
| Dealt ace-king (any suits) | 16/1,326 | 1.2% (1 in 83) |
| Dealt two suited cards | 12/51 | 23.5% |
On the flop
| Event | Probability |
|---|---|
| Pocket pair flops a set or better | 11.8% (about 1 in 8.5) |
| Ace-king (unpaired) pairs at least one card on the flop | 32.4% |
The set figure is the basis of "set mining". If you call a raise with a small pair hoping to flop a set, you miss about 88% of the time. The call is only worth it if the rare hit wins enough to make up for all the misses.
Completing a draw
After the flop there are 47 unseen cards. An out is a card that completes your hand.
| Draw | Outs | Hit on turn | Hit by river | Rule of 4 |
|---|---|---|---|---|
| Flush draw | 9 | 19.1% | 35.0% | 36% |
| Open-ended straight draw | 8 | 17.0% | 31.5% | 32% |
| Gutshot straight draw | 4 | 8.5% | 16.5% | 16% |
The rule of 4 and 2: on the flop, multiply your outs by 4 to estimate your chance of hitting by the river. On the turn, multiply by 2 for one card to come. As the table shows, the shortcut is within a percentage point or so for common draws. It overestimates more as the number of outs grows: with 15 outs it says 60%, against a true 54%.
The exact calculation is the complement of missing twice. For a flush draw that is 1 − (38/47 × 37/46) = 35.0%.
Pot odds
Pot odds turn equity into a decision. If calling costs C and the pot (including your opponent's bet) is P, you need to win more than
C ÷ (P + C) of the time for a call to break even.
Example: the pot is 100, your opponent bets 50, so the pot is now 150 and it costs you 50 to call. You need 50 ÷ 200 = 25% equity. With a flush draw and one card to come (19.1%), the call loses money on this street alone. With two cards to come and no further betting (35%), it would be profitable. Real hands sit between these cases, and implied odds, meaning what you expect to win on later streets when you hit, are what make many draws playable.
Equity against a range
Serious players think in ranges, not single hands: the full set of hands an opponent could plausibly hold. Preflop, pocket aces against pocket kings wins roughly 82% of the time. Against a wide range of hands, aces do even better. Equity calculators enumerate every possible board to compute these figures, which is itself a simple Monte Carlo or exhaustive-enumeration exercise.
Why this matters
In the casino games covered in our house edge guide, the maths is fixed against the player. In poker the maths is shared: everyone faces the same probabilities, and the edge goes to whoever uses them better. The house takes its fee as a rake, a small percentage of each pot, instead.
- Poker probability, Wikipedia
- Pot odds, Wikipedia
- Expected value, Wikipedia