probability
Poker probability.
Three tables. What you get before the flop, out of 1,326 possible starting hands. What the flop does to it, out of 19,600 possible flops. And how often a draw gets there, with the shortcut everyone uses at the table shown next to the real answer so you can see where it starts lying.
Every figure here is counted when this page is built. The flop percentages come from dealing all 19,600 flops and sorting them, not from a simulation and not from a rounded number in a book.
before the flop
What you get, out of 1,326.
Two cards from 52 gives 1,326 combinations. Ignore which suit is which and it collapses to 169 distinct hands, which is the grid on the starting hand chart. Both counts are below, because a hand you hold four ways is four times as common as a hand you hold once.
| Starting hand | Combinations | Chance | Odds |
|---|---|---|---|
| A pocket pair, any pair Two cards of the same rank. | 78 | 5.88% | 1 in 17.0 |
| A specific pair, aces for example Six ways to make it: four suits taken two at a time. | 6 | 0.452% | 1 in 221 |
| Two suited cards Same suit, any two ranks. | 312 | 23.53% | 1 in 4.3 |
| Suited connectors Same suit and neighbouring ranks, from 32s up to AKs. | 48 | 3.62% | 1 in 27.6 |
| At least one ace An ace with anything, pocket aces included. | 198 | 14.93% | 1 in 6.7 |
| Two broadway cards Both cards ten or higher. | 190 | 14.33% | 1 in 7.0 |
| Ace king, either version Twelve offsuit and four suited. | 16 | 1.21% | 1 in 82.9 |
| Ace king suited One for each suit. | 4 | 0.302% | 1 in 332 |
| No pair, no ace, nothing suited or connected The hands you fold without thinking. | 624 | 47.06% | 1 in 2.1 |
78 paired, 312 suited, 936 the rest, and those three add up to 1,326
the flop
19,600 flops, dealt and sorted.
Four example hands, one for each kind of starting hand people actually play. For each of them the build deals every one of the 19,600 flops that can come behind those two cards and counts what happened. Nothing is sampled, so these are not estimates.
The definitions, so the numbers mean something. A pair or better means one of your own ranks appears on the flop. A flush draw means exactly four cards of one suit between your hand and the flop, with at least one of them yours. A backdoor draw means three. A draw with eight outs or more counts the cards left in the deck that would finish a straight, rather than matching a pattern, which is why a double gutshot lands in the same row as an open ender.
No pair and no real draw is the strictest definition here, and it is exactly the same one in both blocks that use it: none of your own ranks on the flop, no made straight, no four to a suit and no four straight outs either. So a gutshot does not count as nothing. That matters: with two unpaired offsuit cards there are 1,880 flops that pair nothing but do leave a gutshot, and they fall outside the last row below rather than inside it.
A pocket pair
| Flop a set, quads or a full house | 2,304 | 11.76% |
| Flop a full house or better | 240 | 1.22% |
| Flop nothing better than the pair you started with | 17,296 | 88.24% |
Two suited cards
| Flop a flush | 165 | 0.842% |
| Flop a flush draw, four to the suit | 2,145 | 10.94% |
| Flop a backdoor flush draw, three to the suit | 8,151 | 41.59% |
| Flop a pair or better | 6,356 | 32.43% |
Two unpaired offsuit cards
| Flop a pair or better | 6,356 | 32.43% |
| Flop two pair | 396 | 2.02% |
| Flop trips | 284 | 1.45% |
| Flop no pair and no real draw | 10,972 | 55.98% |
Suited connectors
| Flop a straight | 256 | 1.31% |
| Flop a draw with eight outs or more | 1,904 | 9.71% |
| Flop a gutshot, four to seven outs | 3,296 | 16.82% |
| Flop a flush draw | 2,145 | 10.94% |
| Flop a pair or better | 6,356 | 32.43% |
| Flop no pair and no real draw | 7,376 | 37.63% |
each set of counts comes from all 19,600 flops, and the build stops if any of them fails to come out at 19,600
drawing
The shortcut, and what it costs.
With one card to come the sum depends on where you are standing. On the flop you have seen five cards, so 47 are unknown and the chance is your outs divided by 47. On the turn 46 remain, and then it is your outs divided by 46. To know from the flop how often you get there by the river, you work out how often you miss both cards, 47 minus your outs over 47 times 46 minus your outs over 46, and subtract that from one. The rule of 2 and 4 replaces those sums with a multiplication, and the columns beside them are the price of that convenience.
So the first two columns belong to different streets, and that is exactly why a flush draw is 19.1 percent in one place and 19.6 percent in another. It is not the same number twice with a rounding difference, it is the turn and the river.
| Draw | Outs | Turn, from the flop | River, from the turn | Outs x 2 | Both cards, from the flop | Outs x 4 | Off by |
|---|---|---|---|---|---|---|---|
| Pocket pair looking for a set | 2 | 4.3% | 4.3% | 4% | 8.4% | 8% | -0.4 |
| Gutshot straight draw | 4 | 8.5% | 8.7% | 8% | 16.5% | 16% | -0.5 |
| Two overcards | 6 | 12.8% | 13.0% | 12% | 24.1% | 24% | -0.1 |
| Open ended straight draw | 8 | 17.0% | 17.4% | 16% | 31.5% | 32% | +0.5 |
| Flush draw | 9 | 19.1% | 19.6% | 18% | 35.0% | 36% | +1.0 |
| Flush draw plus a gutshot | 12 | 25.5% | 26.1% | 24% | 45.0% | 48% | +3.0 |
| Flush draw plus an open ender | 15 | 31.9% | 32.6% | 30% | 54.1% | 60% | +5.9 |
turn is outs divided by 47, river is outs divided by 46, both cards is one minus the chance you miss them both
Read the last column as how much the shortcut flatters you with two cards to come. At 15 outs it is 5.9 points, and that is not a rounding error, it is the gap between a call and a fold on the biggest pot of the night. The shortcut is least reliable exactly where the money is, which is the whole argument for doing the real sum.
the deck, in numbers
| Starting hands you can be dealt | 1,326 |
| Starting hands ignoring suits | 169 |
| Flops, once you hold two cards | 19,600 |
| Turn and river runouts, once the flop is down | 1,081 |
| Complete boards behind your two cards | 2,118,760 |
| Five card hands in a 52 card deck | 2,598,960 |
| Seven card combinations in a 52 card deck | 133,784,560 |
what to do with this
A probability is only half a decision.
Knowing that a flush draw arrives 35.0 percent of the time from the flop through the river tells you nothing on its own. It becomes a decision when you put it next to the price: what you have to pay, and what you win if you get there. That comparison is what the pot odds calculator does, and it is the single most useful sum in the game.
These tables also say nothing about what your opponent holds. A pair of kings is a monster against one player and a liability against three, because every extra player is another chance that someone was dealt the hand that beats you. Running two hands against each other is a different calculation entirely, and it is the one that decides whether you are ahead right now.
For the other side of the deck, how rare each finished hand is rather than how often you reach it, the hand odds page counts every five and seven card combination. It is where the royal flush numbers live, and it explains why the ranking is ordered the way it is. If you want the order itself first, the hand rankings are the place to start.
And none of it replaces knowing the game. How the betting rounds work, who acts first and what a raise obliges you to do are rules rather than arithmetic, and a player who knows the rules cold makes better use of a rough estimate than a player who knows the percentages and misreads the spot.
next
The hard part is counting your outs.
Every percentage in the drawing table starts with a number of outs, and that is the step people get wrong. Cards that make you the second best hand are not outs, and cards your opponent needs are worth less than they look. The calculator counts them with you and then does the arithmetic exactly.
probability questions
What are the odds of being dealt pocket aces?
Six of the 1,326 possible starting hands are pocket aces, so 1 in 221, or 0.452 percent. Any pocket pair at all is 78 of 1,326, which is 1 in 17.0.
How often do you flop a set with a pocket pair?
Counting all 19,600 flops behind a pocket pair, 2,304 of them bring at least a third card of your rank. That is 11.76 percent, or about 1 in 8.5. It is the number behind the old advice about set mining, and it is why you need to win a lot when it lands.
How often do you flop a flush draw with suited cards?
With two suited cards, 2,145 of the 19,600 flops give you exactly four to the flush, which is 10.94 percent. The finished flush comes straight away 0.842 percent of the time, and a backdoor draw, three to the flush, another 41.59 percent.
What is the rule of 2 and 4, and how wrong is it?
Multiply your outs by two with one card to come and by four with two. With one card it always reads low, and furthest on the river from the turn, where you divide by 46 instead of 47: at worst by 2.6 points in this table. With two cards it starts close and then runs away upward: at 15 outs it claims 60 percent while the deck gives 54.1 percent, a gap of 5.9 points. That is the difference between calling a big all in and folding it.
How many possible flops are there?
Once you hold two cards there are 19,600 flops that can come, because 50 cards remain and the order they arrive in does not matter. Behind those, 1,081 turn and river runouts, and 2,118,760 complete boards in total.
How often does a hand improve past one pair?
Not as often as people expect. With two unpaired cards, 32.43 percent of flops pair one of them, but only 2.02 percent give two pair and 1.45 percent give trips. Most of the time the flop misses you completely, which is exactly why position and betting matter more than card strength.
Back to all poker charts.