Odds someone has a bigger pair
I have kings: how likely is it someone has aces? The short answer is below, and under it the working: the formula with the numbers put in, and the count from dealing every possible card.
The working
With your two cards out, 50 remain and all four aces are among them. One named opponent is dealt aces with probability C(4, 2) / C(50, 2) = 6 / 1,225. Two named opponents can both have aces only one way (the remaining two), so add the eight players and subtract the overlap (inclusion–exclusion):
P(one named player has AA) = 6 / 1,225 = 0.490% P(two named players both have AA) = 6 / (1,225 × 1,128) = 0.0004% P(someone of 8 has AA) = 8 × 0.490% − 28 × 0.0004% = 3.906%
For a smaller pair the same sum runs over every higher rank. Each rank of four cards can give one opponent a pair (6 ways) or two opponents a pair each (3 ways), and the build multiplies those out exactly. A seeded simulation of 400,000 full deals per row checks the 8-opponent column; it agrees within sampling error for every pair (for kings it found 3.91%).
Every pair, every table size
Chance that at least one opponent was dealt a higher pocket pair. Columns are the number of opponents: 1 is heads-up, 5 is a 6-max table, 8 is 9-handed.
| Your pair | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| KK | 0.5% | 1.0% | 1.5% | 2.0% | 2.4% | 2.9% | 3.4% | 3.9% | 4.4% |
| 1.0% | 2.0% | 2.9% | 3.9% | 4.8% | 5.8% | 6.7% | 7.7% | 8.6% | |
| JJ | 1.5% | 2.9% | 4.4% | 5.8% | 7.2% | 8.6% | 9.9% | 11.3% | 12.6% |
| TT | 2.0% | 3.9% | 5.8% | 7.6% | 9.5% | 11.3% | 13.0% | 14.8% | 16.5% |
| 99 | 2.4% | 4.8% | 7.2% | 9.5% | 11.7% | 13.9% | 16.0% | 18.1% | 20.2% |
| 88 | 2.9% | 5.8% | 8.6% | 11.3% | 13.9% | 16.5% | 18.9% | 21.4% | 23.7% |
| 77 | 3.4% | 6.7% | 10.0% | 13.1% | 16.1% | 19.0% | 21.8% | 24.5% | 27.1% |
| 66 | 3.9% | 7.7% | 11.3% | 14.8% | 18.2% | 21.4% | 24.5% | 27.5% | 30.3% |
| 55 | 4.4% | 8.6% | 12.7% | 16.5% | 20.2% | 23.7% | 27.1% | 30.3% | 33.4% |
| 44 | 4.9% | 9.6% | 14.0% | 18.2% | 22.2% | 26.0% | 29.7% | 33.1% | 36.4% |
| 33 | 5.4% | 10.5% | 15.3% | 19.9% | 24.2% | 28.3% | 32.1% | 35.8% | 39.2% |
| 22 | 5.9% | 11.4% | 16.6% | 21.5% | 26.1% | 30.4% | 34.5% | 38.3% | 41.9% |
| AK (aces only) | 0.2% | 0.5% | 0.7% | 1.0% | 1.2% | 1.5% | 1.7% | 2.0% | 2.2% |
Queens run into a bigger pair 7.7% of the time 9-handed and jacks 11.3%; deuces, with every pair above them, 38.3%. At 6-max (5 opponents) kings meet aces 2.4% of the time.
What this does not tell you
These are the odds before anyone acts. Betting is information: if a tight player four-bets, the chance they hold aces is far higher than 3.9%, and the table cannot see that. The deal-time figure is still the right starting point, and it is why folding kings preflop at a full table is almost never right. For the price of getting it in against aces anyway, see AA vs KK.
More before the flop
Other odds
- Flopping a set11.8%
- Flopping a flush or flush draw0.8%
- Flopping a straight or straight draw1.3%
- Flopping a pair or two pair32.4%
- An overcard to your pair41.4%
- Flush draw (9 outs)35.0%
- Open-ended straight draw (8 outs)31.5%
- Gutshot (4 outs)16.5%
- Combo draw (12 to 15 outs)54.1%
- Two overcards (6 outs)24.1%
- Set to full house33.4%
- The rule of 2 and 4, checkedup to 12.5 pts off
- Counted, not quoted.
tools/build-odds.mjsdeals every possible flop, turn, river or starting hand and counts. Where a closed-form formula exists, the build refuses to publish unless the count matches it exactly. - Made hands come from the engine. Whether cards make a straight, a flush or a full house is decided by the same Rust/WebAssembly evaluator the trainer runs, not by hand-written rules.
- Odds are not equity. Hitting your hand is not winning the pot. For that, the matchup pages deal every board against a specific hand and the spot calculator works against a range.
- Raw data: odds.json.