Poker Probability — Royal Flush, Quads, Flush and Straight Odds

Probability, odds and outs — three different things

A probability is a percentage chance an event happens (a flush draw completes about 35% of the time from flop to river). Odds express the same thing as a ratio of failures to successes (roughly 1.86-to-1 against). Outs are the number of unseen cards that improve your hand to a likely winner. The quick shortcut most players use is the 2-and-4 rule: multiply your outs by 2 for one card to come, or by 4 on the flop when you will see both the turn and the river. It is an approximation — the exact numbers below are the real figures.

Five-card hand probabilities (dealt from a full 52-card deck)

These are the classic rarity figures for a random five-card hand. They describe how rare each hand is in the abstract, not how often you make it in Texas Hold'em (where you see seven cards).

• Royal flush — 4 combinations, 0.000154% (1 in 649,740)
• Straight flush (including royal) — 40 combinations, 0.00154% (1 in 64,974)
• Four of a kind — 624 combinations, 0.0240% (1 in 4,165)
• Full house — 3,744 combinations, 0.1441% (1 in 694)
• Flush (including straight flushes) — 5,148 combinations, 0.1981% (1 in 505)
• Straight (including straight flushes) — 10,200 combinations, 0.3925% (1 in 255)
• Three of a kind — 54,912 combinations, 2.113%
• Two pair — 123,552 combinations, 4.754%
• One pair — 1,098,240 combinations, 42.257%

Starting hand probabilities in Texas Hold'em

There are 1,326 possible two-card starting combinations. That gives the following exact frequencies before any card is exposed:

• A specific pocket pair such as aces — 6 of 1,326 = 0.452% (1 in 221)
• Any pocket pair — 78 of 1,326 = 5.882% (1 in 17)
• Ace-king (suited or offsuit) — 16 of 1,326 = 1.207%
• Ace-king suited specifically — 4 of 1,326 = 0.302%

Flop probabilities from a Hold'em starting hand

These are exact figures for the flop only, calculated over all 19,600 possible flops given your two known cards.

• Flopping a set or better with a pocket pair — 11.76%
• Flopping quads with a pocket pair — 0.245%
• Flopping a full house with a pocket pair — 0.980%
• Flopping a flush with two suited cards — 0.842%
• Flopping a four-card flush draw with two suited cards — 10.94%
• Flopping a backdoor (three-card) flush with two suited cards — 41.59%
• Flopping a straight with a suited connector such as JT — 1.31% (four qualifying straights: AKQ, KQ9, Q98 and 987, so 4 x 64 = 256 of 19,600 flops)
• Flopping a straight flush with a suited connector such as JT — 0.0204%
• Flopping two pair using both hole cards with an unpaired hand such as AK — 2.02% (3 aces x 3 kings x 44 blanks = 396 of 19,600 flops)
• Flopping a full house with an unpaired hand such as AK — 0.0918%

Drawing odds after the flop

Once the flop is out, 47 cards remain unseen from your perspective. These are the exact completion percentages for the most common draws.

• Flush draw (9 outs), flop to turn — 19.15%
• Flush draw (9 outs), turn to river — 19.57%
• Flush draw (9 outs), flop to river — 34.97%
• Open-ended straight draw (8 outs), flop to river — 31.45%
• Gutshot straight draw (4 outs), flop to river — 16.47%
• One card to come, turn to river (46 unseen): 9 outs — 19.57%; 8 outs — 17.39%; 4 outs — 8.70%

Reading the flop-to-river numbers correctly

The two-card figure only applies if you will actually see both cards. If you call a flop bet and then face another bet on the turn, you are effectively paying for one card at a time, so the relevant number is the one-card figure (about 19.6% for a flush draw), not 35%. Using the two-card number to justify a flop call you cannot afford to follow through on is one of the most expensive misreadings of poker odds.

Turning probability into a decision: pot odds

Compare the price you are being offered with your chance of improving. If the pot is 100 and you must call 25, you are risking 25 to win 125, so you need to win about 20% of the time to break even. A nine-out flush draw with one card to come is 19.15% on the turn and 19.57% on the river, which makes that call roughly break-even on immediate odds — the decision is then settled by implied odds (how much more you can win when you hit) and by whether your opponent can fold.