Let X be the number of heads in 20 fair coin tosses, Y a number picked uniformly at random from {0, 1, . . . 20}, independently of X. Let Z = max(X, Y).
a) Find a formula for P(Z = k), k = 0, . . 20.
b) Without calculating out P(Z = k) exactly, sketch the histogram of Z, and explain its unusual shape.