Question

An urn has n white and m black balls that are removed one at a time in a randomly chosen order. Find the expected number of instances in which a white ball is immediately followed by a black one.

Answer 1

Answer:

Step-by-step explanation:Label the white balls w1 to wn. Let random variable Xi be equal to 1 if white ball wi is immediately followed by a black ball, and 0 otherwise. Then the number Y of instances in which a white is immediately followed by a black is given by Y=X1+⋯+Xn.

By the linearity of expectation we have E(Y)=∑n1E(Xi)=nE(X1).

To find E(X1), we find Pr(X1=1). We have X1=1 if white ball w1 is not chosen last, and is followed by a black. The probability it is not last is m+n−1m+n. Given that it is not last, the probability it is followed by a black is mm+n−1. It follows that Pr(X1=1)=mm+n.

Thus E(X1)=mm+n and E(Y)=mnm+n.