A 5-card hand is dealt from a perfectly shuffled deck. Define the events: A: the hand is a four of a kind (all four cards of one rank plus a 5th card). B: at least one of the cards in the hand is an ace Are the events A and B independent? Prove your answer by showing that one of the conditions for independence is either true or false.

In a hand of 5 cards, you want 4 of them to be of the same rank, and the fifth can be any of the remaining 48 cards. So if the rank of the 4-of-a-kind is fixed, there are possible hands. To account for any choice of rank, we choose 1 of the 13 possible ranks and multiply this count by . So there are 624 possible hands containing a 4-of-a-kind. Hence A occurs with probability

There are 4 aces in the deck. If exactly 1 occurs in the hand, the remaining 4 cards can be any of the remaining 48 non-ace cards, contributing possible hands. Exactly 2 aces are drawn in hands. And so on. This gives a total of

possible hands containing at least 1 ace, and hence B occurs with probability

The product of these probability is approximately 0.000082.

A and B are independent if the probability of both events occurring simultaneously is the same as the above probability, i.e. . This happens if

The above "sub-events" are mutually exclusive and share no overlap. There are 48 possible non-aces to choose from, so the first sub-event consists of 48 possible hands. There are 12 non-ace 4-of-a-kinds and 4 choices of ace for the fifth card, so the second sub-event has a total of 12*4 = 48 possible hands. So consists of 96 possible hands, which occurs with probability

and so the events A and B are NOT independent.

x -

median = 1.25

range = 12-14.5

interquartile = q12-q14.5

y -

median = 1

range = 15-17

interquartile = q15-q17