Three of a kind #7 of 10
Three cards of one rank and two unrelated cards. Called a set with a pocket pair, trips with one card in hand. It beats two pair and everything below it, and loses to a straight and everything above.
How often, card by card
| Cards dealt | Probability | 1 in | Odds against | Count |
|---|---|---|---|---|
| 5 cards: the flop, or five-card draw | 2.11% | 47.3 | 46.3 to 1 | 54,912 / 2,598,960 |
| 6 cards: the turn | 3.60% | 27.8 | 26.8 to 1 | 732,160 / 20,358,520 |
| 7 cards: the river in hold'em, or seven-card stud | 4.83% | 20.7 | 19.7 to 1 | 6,461,620 / 133,784,560 |
“Best hand” means exactly this hand and nothing better: a full house also contains a pair, but it counts once, as a full house. These are the odds for any two hole cards; your own starting hand moves them, which is the grid further down.
The working
Three of a kind, five cards: trips rank × its three suits × two other ranks × their suits 13 × C(4,3) × C(12,2) × 4² = 13 × 4 × 66 × 16 = 54,912 enumerated by the engine: 54,912 of 2,598,960 probability: 54,912 / 2,598,960 = 2.11%
There are 858 different three of a kind hands once suits are ignored: 13 × C(12,2) = 858: the trips rank, then two kicker ranks. With seven cards only 575 of them can be anyone's best hand. The weakest five-card three of a kind is 2♣2♦2♥4♣3♣, but in hold'em the weakest you can finish the river with is 2♣2♦2♥7♦5♣: any weaker seven cards contain something better.
When two players both have three of a kind
Higher three of a kind wins. Same rank (only possible with community cards): the two kickers decide.
Same three of a kind: two kickers play
| A♥2♦ | 6♣6♦6♠A♥J♥ | Three of a kind, sixes, A-J kickers | Wins |
| K♣Q♦ | 6♣6♦6♠K♣Q♦ | Three of a kind, sixes, K-Q kickers | Loses |
Check your own showdown in the hand checker →
Three of a kind by starting hand
The chance each of the 169 starting hands ends the river with three of a kind as its best hand, from every one of its 2,118,760 boards. 3 hands tie for most often (11.8%): AA, KK, 22; 7 hands tie for least often (4.26%): JTs, T9s, 98s and 4 more. Averaged over their combinations: pocket pairs 11.7%, suited hands 4.33%, offsuit hands 4.42%.
| Starting hand | Three of a kind by the river | 1 in |
|---|---|---|
| AA | 11.8% | 8 |
| KK | 11.8% | 8 |
| 22 | 11.8% | 8 |
| 11.7% | 9 | |
| 33 | 11.7% | 9 |
| JJ | 11.7% | 9 |
| 44 | 11.7% | 9 |
| TT | 11.7% | 9 |
| 99 | 11.7% | 9 |
| 88 | 11.7% | 9 |
| … | ||
| 54s | 4.26% | 23 |
| 65s | 4.26% | 23 |
| 76s | 4.26% | 23 |
Three of a kind beats
Three of a kind loses to
- Royal flush 0.0032% by the river
- Straight flush 0.028% by the river
- Four of a kind 0.17% by the river
- Full house 2.60% by the river
- Flush 3.03% by the river
- Straight 4.62% by the river
Making three of a kind: the odds pages
- Every five-, six- and seven-card hand was dealt and classified by the site's engine (
tools/build-rankings.mjs). The five-card count must equal the formula above or the build fails; the six- and seven-card counts must equal the standard published tables. - The starting-hand figures deal each hand every possible board. Suits are interchangeable, so one deal of each hand gives the exact figure for all its combinations.
- The showdowns above were scored by the same engine; the build fails if any of them comes out differently from what its title says.
- Back to all ten hands. Raw data: rankings.json.