1. Let X be binomially distributed with parameters n and p. Show that as k goes from 0 to n, P(X = k) increases monotonically, then decreases monotonically reaching its largest value

(a) in the case that (n + 1)p is an integer, when k equals either (n + 1)p - 1 or (n + 1)p,

(b) in the case that (n + 1)p is not an integer, when k satis?es (n + 1)p - 1 k (n + 1)p.

2. An airline knows that 5 percent of the people making reservations on a certain ?ight will not show up. Consequently, their policy is to sell 52 tickets for a ?ight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?