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.

step-by-step explanation:

you can add the two numbers and divide the result by 2. or you can find the gap between the two numbers and add half of this to the smaller number.