If there is no winner for the lottery described in Problem 2.7.8, then the pot is carried over to the next week. Suppose that in a given week, an r dollar pot is carried over from the previous week and 2n tickets sold. Answer the following questions.
(a) What is the probability q that a random ticket will be a winner?
(b) If you own one of the 2n tickets sold, what is the mean of V, the value (i.e., the amount you win) of that ticket? Is it ever possible that E[V] > 1?
(c) Suppose that in the instant before the ticket sales are stopped, you are given the opportunity to buy one of each possible ticket. For what values (if any) of n and r should you do it?
Problem 2.7.8
In the New Jersey state lottery, each $1 ticket has six randomly marked numbers out of 1,..., 46. A ticket is a winner if the six marked numbers match six numbers drawn at random at the end of a week. For each ticket sold, 50 cents is added to the pot for the winners. If there are k winning tickets, the pot is divided equally among the k winners. Suppose you bought a winning ticket in a week in which 2n tickets are sold and the pot is n dollars.
(a) What is the probability q that a random ticket will be a winner?
(b) What is the PMF of Kn, the number of other (besides your own) winning tickets?
(c) What is the expected value of Wn, the prize you collect for your winning ticket?