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Probability calculator.
Two questions and two formulas. What are the odds of drawing the cards you need out of the ones you have not seen, and what are the odds of a rare hand turning up at least once over a long run. Both are exact. Nothing here is simulated or rounded to a number that sounds right.
the calculator
47 on the flop, 46 on the turn
30,940 is a royal flush, 221 is pocket aces
At least once
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Not at all
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As odds
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| How many arrive | Chance |
|---|
the same numbers, spelled out
A flush draw on the flop is in the fields: 47 cards you cannot see, 9 that make your hand, 2 still to come.
drawing from the deck
Count the ways, divide by the ways.
The easiest route to the answer is the back way round: work out how often you miss completely, then subtract. With nine outs and two cards to come, you miss the turn 38 times in 47 and then the river 37 times in 46, which multiplies to the chance of missing both. One minus that is 35.0 percent, and that is where the number everyone quotes comes from.
| Situation | Unseen | Outs | To come | You hit |
|---|---|---|---|---|
| Flush draw, both cards to come | 47 | 9 | 2 | 35.0% |
| Flush draw, the river only | 46 | 9 | 1 | 19.6% |
| Open ender, both cards to come | 47 | 8 | 2 | 31.5% |
| Gutshot, the river only | 46 | 4 | 1 | 8.7% |
| Pocket pair, hitting a set on the flop | 50 | 2 | 3 | 11.8% |
| Two overcards pairing by the river | 47 | 6 | 2 | 24.1% |
worked out at build time, and checked against the plain multiplication before the page is allowed to exist
at least once
Rare things over a long run.
If something happens once in a thousand, it does not follow that a thousand tries make it certain. The right sum is one minus the chance of never seeing it, which is one minus p to the power of the number of tries. Multiplying instead double counts every run where it happens twice, and the error grows as the number of tries approaches the odds.
| Hands played | A royal flush, at least once | Pocket aces, at least once |
|---|---|---|
| 100 | 0.323% | 36.5% |
| 1,000 | 3.2% | 98.9% |
| 10,000 | 27.6% | 100.0% |
| 50,000 | 80.1% | 100.0% |
| 100,000 | 96.1% | 100.0% |
A royal flush arrives about once in 30,940 seven card hands, and it takes roughly 21,446 hands before you are more likely to have seen one than not. Pocket aces come once in 221, and the same halfway mark is about 153 hands. Those two numbers are lower than the odds themselves, which surprises people: the median wait for a rare event is shorter than its average wait. The royal flush odds come from the same count as the hand odds page, so the two cannot drift apart.
what a probability is not
A probability is not equity. This page counts how often a card arrives; equity is how often you actually win the pot, which depends on what the other player holds and on whether the card that turns up is genuinely good for you. A heart that completes your flush and pairs the board at the same time is a hit and a loss. Counting outs honestly, including the ones that are not clean, is where that distinction gets handled.
A probability is also not a decision. Whether to pay for a draw depends on the price, and comparing the chance against the price is the sum that decides it. Twenty percent is a fold against one bet and a call against another.
And a probability says nothing about a run being due. The deck has no memory, so a thousand hands without aces changes nothing about the next one. If you want the standing figures rather than one calculation, the probability tables count every flop for a set of example hands, and if you want to know how rare each finished hand is, the hand rankings put the order and the counts side by side.
next
The odds for every hand, already counted.
This calculator answers one question at a time. The hand odds page answers all of them at once: every poker hand with its exact count, for five cards and for seven, including the royal flush numbers this page uses.
probability questions
How do you calculate poker probability?
Count the outcomes you want and divide by all the outcomes there are. For cards still to come that is the hypergeometric rule: the number of ways to draw the cards you need, times the ways to fill the rest, over the total number of draws. It is exact, it needs no simulation, and it is what this page does.
What is the chance of hitting a flush draw?
With nine outs and two cards to come it is 35.0 percent, and with one card to come 19.6 percent. Both come from the deck rather than from the rule of 2 and 4, which would say 36 and 18. The shortcut is close on this particular draw and drifts badly on bigger ones.
How often will I see a royal flush?
A seven card hand contains one about once in 30,940 hands. Over 10,000 hands the chance of seeing at least one is 27.6 percent, and it takes roughly 21,446 hands before that chance passes half. The reason it is lower than you expect is that the same rare event can happen twice in a run, so the chances do not simply add up.
Why can I not just multiply the odds by the number of hands?
Because that counts the runs where it happens twice as if they were two separate successes. The right sum is one minus the chance of never seeing it: 1 minus (1 minus p) to the power n. Over 1,000 hands, multiplying would claim 452.5 percent for pocket aces while the real answer is 98.9 percent.
What are the odds of pocket aces?
Six of the 1,326 starting hands are pocket aces, so 1 in 221. You get one at least once in a session of 100 hands about 36.5 percent of the time, and the run of hands before a fifty percent chance is around 153.
Is this the same as equity?
No, and the difference matters. This works out how often a card or a hand appears. Equity is how often you win the pot, which also depends on what your opponent holds and on whether the card that arrives is actually good for you. A card that completes your flush and gives someone a full house is a hit and not a win.
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