M3 Written Expected Value Suppose you shuffle a deck of cards thoroughly and then select cards from the deck (one at a time) without replacement until the ace of hearts appears

M3 Written Expected Value

Suppose you shuffle a deck of cards thoroughly and then select cards from the deck (one at a time) without replacement until the ace of hearts appears.

1. Let A_k be the event that the ace of hearts is drawn on the kth draw. Use a tree diagram to show that

                                                                      P(A_k) =

,           k = 1,2,...,52

by illustrating the first five draws. That is, the probability that you draw the ace of hearts on any given draw is always

. I’ve started the tree diagram for you. Start

                                                                                 not A♥                     A♥

1

2. Let X be the random variable representing the number of cards it takes to draw the ace of hearts. Find the expected value of X to determine the number of cards one would expect to draw to get the ace of hearts. In your calculation, you may find this identity useful:

n

X n(n + 1) k =

                                                                                 k=1                       2