What is the chance of flopping a full house?
With a pocket pair there are two ways. One more card of the pair's rank can arrive with a pair of another rank on the flop (144 flops), or the flop can be three of one other rank (48 flops). Together that is 192 of 19,600 flops, 0.98%.
| Event | Chance | Odds | Flops out of 19,600 |
|---|---|---|---|
| Pocket pair: a set plus a pair on the board | 0.73% | 1 in 136 | 144 |
| Pocket pair: three of another rank on the board | 0.24% | 1 in 408 | 48 |
| Pocket pair: any full house | 0.98% | 1 in 102 | 192 |
| Unpaired hand: any full house | 0.09% | 1 in 1,089 | 18 |
How can an unpaired hand flop a full house?
The flop must pair both hole ranks with one card repeated: two cards of one rank and one of the other. That is 2 x C(3, 2) x 3 = 18 flops, 0.092%. A board of three cards of one rank gives an unpaired hand only trips on the board, which every player shares.
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 full house with a pocket pair?
- 0.98%, or 1 in 102.
- What is the probability of flopping a full house with an unpaired hand?
- 0.092%, or 1 in 1,089.
- Is a full house on the flop common?
- No. Even with a pocket pair it comes 1 in 102, which is about once in every 102 deals of that pair.