Starting Hand Odds: How Likely Are Aces, Pairs and Ace-King?
Counting the 1,326 possible starting hands shows how rare pocket aces are and how common pairs and suited cards really are.
Every poker conversation seems to include a story about getting dealt pocket aces three times in a row, or never seeing a good hand all night. Counting the possible starting hands shows how much of this is just probability at work. The calculations below are simple, and they make a good introduction to how combinations behave in Texas Hold'em.
How many starting hands exist?
A standard deck has 52 cards, and you receive two of them. The number of different two-card combinations is 52 × 51 ÷ 2, which equals 1,326. Order does not matter, so ace of spades followed by king of hearts is the same hand as king of hearts followed by ace of spades. Every probability below is a count of combinations divided by 1,326.
Pocket aces
There are four aces in the deck, and the number of ways to choose two of them is 4 × 3 ÷ 2 = 6. So the probability of being dealt pocket aces is 6 ÷ 1,326, which is about 0.45 percent, or roughly one time in 221 hands. If you play a few hundred hands in an evening, you should not be surprised to see them once or perhaps not at all. The same figure applies to kings, queens and every other specific pair.
Any pocket pair
There are 13 ranks, and each has six possible pairs, so the total is 13 × 6 = 78 combinations. The probability of any pair is 78 ÷ 1,326, about 5.9 percent, or roughly one hand in 17. That is much more common than a specific pair, which is why pairs are not as special as a single set of aces.
Ace-king
There are four aces and four kings, so the number of ace-king combinations is 4 × 4 = 16. The probability of being dealt any ace-king is 16 ÷ 1,326, about 1.2 percent, or roughly one in 83. Of those 16 combinations, four are suited, which gives about 0.3 percent for a specific suited ace-king.
Suited hands
For a suited hand, you choose a suit (four ways) and then two cards of that suit, which is 13 × 12 ÷ 2 = 78 ways. That gives 4 × 78 = 312 suited combinations. The probability of being dealt two cards of the same suit is 312 ÷ 1,326, about 23.5 percent, or a little under one hand in four.
Holding at least one ace
The easiest way to count this is to subtract the hands with no ace. There are 48 non-aces, so the number of two-card hands without an ace is 48 × 47 ÷ 2 = 1,128. The hands with at least one ace number 1,326 minus 1,128, which is 198. So the probability is 198 ÷ 1,326, about 14.9 percent, or roughly one hand in seven.
What the numbers tell you
Premium hands are rare, and most of the time you will hold something ordinary. This is part of why patience matters in the game, and also why streaks of weak hands are normal. It is also a reminder that a good starting hand does not decide the result. Even aces lose fairly often, as explained in our look at variance in poker. And the way hands improve after the flop is covered in our guide to outs and the rule of 4 and 2.
Keep it in perspective
Knowing the odds is interesting, but it does not remove the risk of losing money. If you play for stakes, set a budget in advance and play only if you are of legal age where you live.
Frequently asked questions
Why is the total 1,326 and not 2,652? Because the order of the two cards does not matter. Counting both orders would double every hand, so we divide by two.
Are suited hands much stronger than unsuited ones? Being suited adds some potential for a flush, but the effect is modest compared with the rank of the cards.
Is any specific pair equally likely? Yes. Each pair has six combinations, so each has the same chance, about 0.45 percent.