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Each card can be chosen only once, because a real deck has no duplicates. Select a chosen card again to remove it.
No cards chosen yet.
The same seed and settings always reproduce exactly the same results.
Draw number 0. Reusing the same seed repeats the same sequence of draws.
These counts come from combinatorial counting arguments (explained in the combinations lesson) and are verified in this project's tests by checking every one of the 2,598,960 combinations. Bars use a logarithmic length so rare patterns stay visible.
| Pattern | Example | Combinations | Probability | Frequency | Relative size (log scale) |
|---|---|---|---|---|---|
| Straight flush | 40 | 0.0015% | 1 in 64,974 | ||
| Four of a kind | 624 | 0.0240% | 1 in 4,165 | ||
| Full house | 3,744 | 0.1441% | 1 in 694 | ||
| Flush | 5,108 | 0.1965% | 1 in 509 | ||
| Straight | 10,200 | 0.3925% | 1 in 255 | ||
| Three of a kind | 54,912 | 2.1128% | 1 in 47 | ||
| Two pair | 123,552 | 4.7539% | 1 in 21 | ||
| One pair | 1,098,240 | 42.2569% | 1 in 2 | ||
| No pattern (high card) | 1,302,540 | 50.1177% | 1 in 2 | ||
| Total | 2,598,960 | 100.00% | |||
See the full hand rankings chart and download the poster →
A combination is a selection in which order does not matter: A♠ K♠ Q♠ J♠ 10♠ is the same combination however you arrange it in your hand. The number of ways to choose five cards from fifty-two is written C(52, 5) and equals 52 × 51 × 50 × 49 × 48 ÷ (5 × 4 × 3 × 2 × 1) = 2,598,960.
The patterns in the table (pairs, runs of consecutive ranks, a single suit and so on) are simply labels for groups of combinations that share a structure. They are useful in mathematics because each group can be counted exactly. For example, there are 13 ranks to choose for a pair, C(4, 2) = 6 suit-pairs for that rank, C(12, 3) = 220 ways to pick the three other distinct ranks and 4 × 4 × 4 suits for them: 13 × 6 × 220 × 64 = 1,098,240 combinations with exactly one pair.
Notice how a precise, specific combination is no more or less likely than any other, yet the kinds of combination have wildly different sizes. That distinction between an individual outcome and an event (a set of outcomes) is the central idea of probability. Continue with the combinations lesson or test these numbers with the hand-frequency simulator.