Breaking ties in five-card poker: high cards and split pots

Two players each show a pair of aces. Neither hand outranks the other by category, so something else has to decide it. In five-card poker that something is a fixed order of comparison: look at the cards that define the hand first, then the leftover cards, highest to lowest. If every rank matches, the hands are equal and the pot is split. Nothing else is consulted, and in particular suits are never used to separate two hands.

The procedure, in one rule

Sort each hand into groups by how many cards share a rank, biggest group first, and within a group by rank, highest first. Then compare the two lists from the left and stop at the first difference. A hand of K-K-4-4-A becomes the list K, 4, A (higher pair, lower pair, spare card). A hand of Q-Q-J-J-K becomes Q, J, K. The first entries are K and Q, so the first hand wins and the rest is never looked at.

The spare cards that come last are called kickers. They matter only when everything before them is equal.

• High card and flush: compare all five cards from the top down. A-J-9-5-3 beats A-J-8-7-6, because the first two cards match and the third decides it (9 against 8). The 7 and 6 in the losing hand never come into it.
• One pair: the pair's rank first, then the three spare cards in order. A-A-K-7-2 beats A-A-Q-J-10 on the third card, even though the second hand has more middling cards.
• Two pair: higher pair, then lower pair, then the single spare card.
• Three of a kind: the rank of the three, then the two spares in order.
• Full house: the three of a kind decides it. 3-3-3-A-A beats 2-2-2-K-K, because the triplet of threes outranks the triplet of twos and the pair is never compared.
• Four of a kind: the rank of the four, then the fifth card.
• Straight and straight flush: only the top card matters, because the other four follow from it.

The straight with a low ace

One straight needs a note. The ace counts high in 10-J-Q-K-A and low in A-2-3-4-5, which is called the wheel. In the wheel the five is the top card, so it is the lowest straight there is and loses to 2-3-4-5-6. The ace does not wrap round, so Q-K-A-2-3 is not a straight at all. Those ten sequences, from five-high to ace-high, are the only ten straights, and each is a straight flush when all five cards share a suit.

Why suits do not count

In standard poker the four suits are equal. A flush of spades and a flush of hearts with the same five ranks are the same hand. That is what makes exact ties possible: the same ranks can appear in different suits, so two players can hold hands that look different and are worth precisely the same.

Here is one that can happen at a real table. One player holds A♠ A♥ K♣ Q♦ J♠ and the other holds A♣ A♦ K♥ Q♠ J♦. The ranks are A, A, K, Q, J in both cases, so the hands tie and each player takes half of the pot.

The Poker vs Dealer page describes a game in which your five cards compete against the dealer's, who builds a hand one card at a time, face up. The listing does not say how an exact tie is settled in the app, so the in-game rules screen is the place to check that. The ordering above is the standard one for five-card hands.

How many different hands are there

Counting distinct strengths gives a useful picture of how much room each category has for ties. Within a category, a hand is defined by its rank pattern alone, since suits do not matter. That gives these counts:

• High card: 1,287 ways to choose five different ranks, minus the 10 that form a straight, leaves 1,277. A flush has the same 1,277 patterns.
• One pair: 13 ranks for the pair and C(12,3) = 220 ways to choose three different spare ranks: 13 × 220 = 2,860.
• Two pair: C(13,2) = 78 choices of pairs and 11 remaining ranks for the spare card: 78 × 11 = 858.
• Three of a kind: 13 ranks and C(12,2) = 66 spare pairs: 13 × 66 = 858.
• Full house: 13 × 12 = 156. Four of a kind is also 13 × 12 = 156.
• Straight and straight flush: 10 each.

Add them up: 1,277 + 2,860 + 858 + 858 + 10 + 1,277 + 156 + 156 + 10 = 7,462. That is the number of distinct hand strengths in five-card poker, from 7-5-4-3-2 at the bottom to a royal flush at the top. Every possible five-card hand, all 2,598,960 of them, falls into exactly one of those 7,462 places, and hands in the same place tie.

What this means for a pair

One pair is the busiest category for tie-breaks. It has 2,860 strengths and 1,098,240 hands, an average of 384 hands for each strength. That is why a pair so often comes down to the third spare card, and why the pair rank alone rarely settles a showdown. Compare that with a straight: 10,200 hands share just 10 strengths, so two straights almost always come down to the top card, with the occasional exact tie when both players hold the same sequence.

The full category counts are in How rare each five-card poker hand is, and Why a flush beats a straight explains why the categories are ordered as they are.