What is the chance of flopping a set with a pocket pair?
A pocket pair has two cards of its rank left among the 50 unseen cards, and it flops a set or better when at least one of them is on the flop. That happens on 2,304 of the 19,600 possible flops, 11.76%, or 1 in 8.51. The odds against are 7.51 to 1. The chance is the same for every pocket pair, from deuces to aces.
| Event | Chance | Odds | Flops out of 19,600 |
|---|---|---|---|
| A set or better (at least one more card of the pair) | 11.76% | 1 in 8.51 | 2,304 |
| A set with an unpaired board (exactly one more card) | 10.78% | 1 in 9.28 | 2,112 |
| A set that pairs the board, a full house | 0.73% | 1 in 136 | 144 |
| Four of a kind (both remaining cards) | 0.24% | 1 in 408 | 48 |
| No card of the pair on the flop | 88.24% | 1 in 1.13 | 17,296 |
How is the chance of flopping a set calculated?
Count the flops that miss and subtract. There are C(48, 3) = 17,296 flops that use none of the two remaining cards, out of C(50, 3) = 19,600. So the chance to flop a set or better is 1 - 17,296 / 19,600 = 11.755%.
What is the chance of making a set by the turn or the river?
The same count runs on more cards. With four cards seen (the flop and the turn) the chance of at least one more card of the pair's rank is 15.51%, 1 in 6.45, and with all five board cards it is 19.18%, 1 in 5.21. A pocket pair that misses the flop still has two more chances.
| Board cards seen | Chance | Odds | Odds against |
|---|---|---|---|
| The flop (3) | 11.76% | 1 in 8.51 | 7.51 to 1 |
| The turn (4) | 15.51% | 1 in 6.45 | 5.45 to 1 |
| The river (5) | 19.18% | 1 in 5.21 | 4.21 to 1 |
What does the set chance mean for set mining?
A set mine is a call with a pocket pair to see a flop. Hitting is 7.51 to 1 against, so you need to win about 7.5 times what you call, on average, when the set comes in, just to break even on the flops you hit and the ones you miss. Opponents who will pay you off with a big hand when you do hit (implied odds) are what make the call worth it.
Where do these numbers get used?
The chance to flop a hand is the first number in a plan. The pages below turn it into prices: what a call can cost, how many outs a draw has, and how many ways a hand can be dealt.
How were these numbers computed?
After two hole cards are dealt, 50 cards are unseen and the flop is any 3 of them, so there are C(50, 3) = 19,600 equally likely flops. For each starting hand we deal all 19,600, classify the five cards with our hand evaluator and count. Suits are symmetric, so one combination stands for every combination of the same hand. The closed forms on this page use binomial coefficients and give the same counts.
Source: GeorgeLiPoker's own exact count of the 52-card deck.
Questions
- What is the probability of flopping a set?
- 11.76%, or 1 in 8.51. That is the chance that at least one of the two remaining cards of a pocket pair's rank is on the flop, and it is the same for every pocket pair.
- What is the chance of hitting a set by the river?
- 19.18%, or 1 in 5.21, counting all five board cards. By the turn it is 15.51%.
- Is a set the same as three of a kind?
- A set is three of a kind made with a pocket pair and one card on the board. Three of a kind made with one card in your hand and a pair on the board is called trips. The chance in this page is for a set.