Odds of four aces in a hand?

Answer 1

Assuming 5 cards are drawn, the probability is 1.84689 * 10-5. The odds are 1 time in 54145 times. This can be expressed as a decimal, 0.0000184689 or as a percentage, 0.00184689% . Also, I checked these numbers against Wikipedia Poker probability. They give the probability of 0.0240%, but this is for any four of a kind. For the ace four of a kind, we have to divide by 13, and get 0.001846%. The difference of 0.00000089% is due to roundoff. Explanation: There are 52 cards in a deck. A draw of 5 cards means the number of combinations is: 52!/(5! x 47!) = 2598960 different hands (order of cards doesn't make a difference) Four of a kind = only one combination can form a four of a kind, but since we have 5 cards, one card is anything but an ace (one of 48 cards). So there are 48 combinations that form a 4 of a kind. 48/ 2598960 = 1.84689 * 10-5 I used excel in solving this. The combinations can easily be solved by +fact(52)/(fact(47)*fact(5)) Of course, in poker you can exchange cards which I have not included in the calculations.