Consider a lottery. Each ticket is 1,000 WON and has six numbers out of {1,2,...,45}. A ticket is a winner if its six numbers all match the six numbers drawn at random at the end of a week (assume partial matchings are not rewarded). For each ticket sold, 500WON is added to the pot for the winners. If there are k winners, the prize money is split equally among the winners. Suppose that you bought a ticket in a week in which the total of 2n tickets were sold.

(a) What is the probability p that a single ticket is a winning ticket?

(b) Suppose that your ticket is a winning ticket. Let Kn be the number of other winning tickets (besides yours). Assume that whether a ticket becomes a winning ticket or is independent of all other tickets. Find the PMF of Kn. (Hint. There are 2n 1 tickets besides yours.)

(c) Again, your ticket is a winning ticket. Let Wn be the prize money you get for your winning ticket. Calculate the expected value of Wn. Simplify the expression as much as you can. (Hint. When there are k other winners, you get 1/k+1 of the total prize money.)

(d) Does the expected prize money increase linearly with the sales of tickets?