* 8. An ambidextrous student has a left and a right pocket, each initially containing n humbugs.

Each time he feels hungry, he puts a hand into one of his pockets and, if it is not empty, he takes a humbug from it and eats it. On each occasion, he is equally likely to choose either the left or right pocket.When he first puts his hand into an empty pocket, the other pocket contains H humbugs.

Show that if ph is the probability that H = h, then ph =



2n − h n



1 22n−h for h = 0, 1, . . . , n, and find the expected value of H, by considering Xn h=0 ph, Xn h=0 hph, Xn h=0

(n − h)ph, or otherwise. (Oxford 1982M)