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

What poker probability actually measures

Every probability in poker is a ratio between two counts: how many card combinations produce the result you care about, and how many combinations are possible at all. Nothing else is involved — no streaks, no "due" cards. A 52-card deck produces C(52,5) = 2,598,960 distinct five-card hands, and every published hand frequency is simply a count divided by that number. This page is the methodology page: how the numbers are derived and how to reproduce them at the table. For the full reference tables see our probability hub, and for flop-specific events see the flop odds page.

Combinations: the only formula you need

Order does not matter in a poker hand, so every count uses the combination formula C(n,k) = n! / (k! × (n−k)!). Three numbers do almost all the work: C(52,5) = 2,598,960 — all five-card hands. C(50,3) = 19,600 — all possible flops once you hold two known cards. C(4,2) = 6, C(4,3) = 4 — the ways to pick two or three suits of one rank. A specific hand is worth remembering: a pair has 6 combinations, a suited hand has 4, an offsuit hand has 12. That is why 169 "hand types" become 1,326 real starting combinations.

Unseen cards, not remaining cards

The most common beginner error is counting the deck instead of counting unseen cards. After the flop you have seen 5 cards (your two plus three board cards). From your point of view 47 cards are unseen — opponents' hole cards are unseen too, so they still count. After the turn, 46 cards are unseen. This matters because every draw calculation divides by 47 or 46, never by "the deck minus the cards that were dealt to everyone".

• Preflop, before you look: 52 unseen
• Preflop, holding two cards: 50 unseen
• On the flop: 47 unseen
• On the turn: 46 unseen

Counting outs correctly

An out is an unseen card that gives you the best hand. Count them by naming them, not by guessing. Flush draw: you hold two hearts, the flop shows two hearts. There are 13 hearts; you can see 4, so 9 hearts remain unseen → 9 outs. Open-ended straight draw: 8-9 in hand, 6-7 on the board. Any 5 or any T completes it → 4 + 4 = 8 outs. Gutshot: 8-9 in hand, 6-T on the board. Only a 7 completes it → 4 outs. Two overcards: A-K on a 7-5-2 board. Three aces plus three kings → 6 outs, and these are "soft" outs because hitting top pair does not always win.

Converting outs into equity — the exact numbers

With O outs and 47 unseen cards, the probability of missing the turn is (47−O)/47, and of missing both turn and river is ((47−O)/47) × ((46−O)/46). Subtract from 1 to get the chance of hitting. Worked example, flush draw with 9 outs: Turn only: 9/47 = 19.1%. Turn or river: 1 − (38/47 × 37/46) = 1 − 0.6503 = 34.97%. Worked example, gutshot with 4 outs: Turn only: 4/47 = 8.5%. Turn or river: 1 − (43/47 × 42/46) = 1 − 0.8353 = 16.47%.

Why the shortcut and the exact number disagree

The rule of 2 and 4 says: multiply outs by 2 for one card, by 4 for two cards. It works because 1/47 ≈ 2.1%. The two-card version is a rough addition of two probabilities and ignores the overlap where you hit on both streets, so it drifts upward as outs increase — accurate within a point up to about 8 outs, roughly 6 points too generous at 15 outs. The second, larger caveat: the ×4 version is only valid when you are guaranteed to see both cards, i.e. you or your opponent are all-in on the flop. If there is another betting round, price your call against the turn card alone (×2) unless you have a concrete plan and the stack depth to see the river.

Flop probabilities from a known starting hand

These are the numbers most players actually use, all derived from C(50,3) = 19,600 possible flops. Pocket pair flopping a set or better: 1 − C(48,3)/C(50,3) = 1 − 17,296/19,600 = 11.76% (about 1 in 8.5). Suited hand flopping a flush: C(11,3)/19,600 = 165/19,600 = 0.84%. Suited hand flopping a flush draw (exactly two more of the suit): C(11,2) × 39 / 19,600 = 2,145/19,600 = 10.94%. Being dealt a pocket pair at all: 3/51 = 5.88%, or 1 in 17. Being dealt A-K (any): 16/1,326 = 1.21%.

Turning equity into a decision: pot odds

Equity alone decides nothing. Compare it to the price you are being asked to pay. Pot odds = call / (pot after your call). If the pot is 100 and your opponent bets 50, you call 50 into a pot that will be 200 including your call — you need 50/200 = 25% equity to break even. Worked example: you hold a flush draw on the flop and face that 50-into-100 bet with money behind. Equity to the turn alone is 19.1%, which is below 25% — a pure call is short. Add the times you win by hitting the river after seeing a free or cheap turn, plus the extra money you win when you do hit (implied odds), and the call becomes defensible. Add the times you can win by betting yourself (fold equity) and it becomes clearly profitable. That is the whole calculation: equity, price, and what happens on later streets.

Common probability mistakes, corrected

Each of these costs real money and each has a one-line correction.

• Counting outs that give an opponent a better hand — a heart completing your flush can complete their straight flush or fill a board pair. Discount "tainted" outs.
• Using the ×4 rule when more betting is coming. It is only an all-in shortcut.
• Dividing by 52 or by "cards left in the deck". Always divide by unseen cards: 47 on the flop, 46 on the turn.
• Treating independent events as connected. Missing five flush draws in a row does not raise the odds on the sixth.
• Adding probabilities instead of subtracting the miss chance. 19.1% + 19.6% is not 35% — the correct method is 1 minus the product of the misses.
• Ignoring the number of opponents. Your equity against one player and against four players are very different numbers for the same hand.