Two Plus Two Older Archives - Odds of a royal flush, and four of a kind

A poker hand is made up of 5 cards, and there are 52 cards in the deck. This means there are 52 ways to pick the first card, 51 ways to pick the second card, and so on. So, there are 52*51*50*49*48 ways to pick a 5-card hand. However, some of those hands are identical (it doesn't matter what order we get the cards in) so we divide to get rid of the duplicates. Mathematically, we want (52 choose 5) which is 52! / (52-5)!5!, which is 52!/47!5!, or 52*51*50*49*48/120. Thus, there are 2,598,960 possible poker hands.

Say we want the odds of four of a kind. There are 13 different quads (2222-AAAA), and for each set of quads, the 5th card can be any of the remaining 48 cards in the deck, which means there are 13*48=624 different ways to get four of a king. Dividing this into the number of possible poker hands, above, shows we can expect quads once in 4165 hands; another way of putting it is the odds against getting quads are 4,164 to 1.

A Royal Flush, on the other hand, has only 4 ways to get it, so the odds are 4/2,598,960, or 649,739 to 1.

Note that that is for a 5-card hand; since hold'em gives us 7 cards, we effectively get (7 choose 5) = 21 possible poker hands each time.