Poker

Poker probability

Poker probability is the calculation of the chance that a specified hand, draw, or outcome will occur in a poker situation. It is expressed as a fraction, decimal, or odds ratio.

Sample space (5-card hand): 2,598,960 possible hands¹
Royal flush probability: 4 in 2,598,960 (≈ 1 in 649,740)¹
Three of a kind probability: ≈ 2.11%¹
One-card draw formula: P = outs ÷ unseen cards²
Two-card draw formula: P = 1 − (non-outs ÷ unseen) × ((non-outs−1) ÷ (unseen−1))²

Key points

Poker probability counts favorable card combinations divided by total possible outcomes.¹
For a flush draw on the turn with 9 outs and 47 unseen cards, the chance is 9/47.³
Cumulative probability gives the chance of a hand at least as good as a specified rank, e.g. ≈ 2.87% for three of a kind or better.¹
The odds against a royal flush are 649,739:1.¹

Flush draw on the turn

You hold four hearts after the flop. There are 13 hearts in the deck; you have seen 4, leaving 9 hearts as outs. With 47 unseen cards, the probability of completing the flush on the turn is 9/47 ≈ 19.1%.³

Common questions

How to calculate poker probability?

For a single card to come, divide the number of outs by the number of unseen cards. For two cards to come, use the formula: 1 − (non‑outs ÷ unseen) × ((non‑outs−1) ÷ (unseen−1)).² For a full five‑card hand, count the number of ways to make the hand and divide by 2,598,960.¹

Where this term is used

Not the same as

Poker hands probability — a narrower term for probabilities of specific hand types, whereas poker probability also covers draws, outs, and win chances¹²
Poker hand rankings probability — the probabilities of ranked hand categories in a random deal, while poker probability is broader and includes live-hand situations¹⁴

Poker Poker hands probability Probabilities of specific five-card poker hand types. Poker Poker hand rankings probability Probabilities of ranked hand categories in a random deal. Poker Full house poker A hand consisting of three of a kind and a pair. Poker Gto poker Game theory optimal strategy for poker. Poker Draw poker A poker variant where players can exchange cards. Poker Quads poker Four of a kind in poker.

Sources

Poker probability en.wikipedia.org Provides the sample space, royal flush probability, three of a kind probability, cumulative probability, and odds against.
Draw (poker) en.wikipedia.org Supplies the one-card and two-card draw probability formulas.
Out (poker) - Wikipedia en.wikipedia.org Gives the flush draw example with 9 outs out of 47 unseen cards.
343 arxiv.org Supports the distinction between poker probability and poker hand rankings probability.

Sources are drawn from regulators, universities and published research, and each one is labelled with what it actually is — a preprint is not called a paper. Bookmaker and affiliate pages are never cited here, because a page that sells betting is not a neutral authority on it.

Written and maintained by Ellis Warrender, writer and maintainer.

Last updated 28 July 2026. Spotted an error? Tell me and it gets fixed and logged on the corrections page.