Answer to Question #439 in Statistics and Probability for Duman

Question

While distributing the deck of 52 cards to four players one of them did not get aces three times in a row. Does he have a reason to complain about bad luck?

Answer 1

Each player receives 52/4 = 13 cards. There are 4 aces and 48 "not aces" in the deck. The number of ways to choose 13 cards out of 52 is (C52)^13. We are considering the situation when among 13 cards there are 0 aces and 13 "not aces". The number of ways to get 13 "not aces" out of the 48 possible cards is (C48)^13.

The probability of such a hand is: P = ((C48)^13)/(C52)^13 = (48!*13!*39!)/(13!*35!*52!) ≈ 0.3.
The probability of such a hand 3 times in a row is : P(3) = P3 ≈ 0,028. As you can see, there was an event with the probability less than 3%, that is, a player is really unlucky.