Each round played by a contestant is either a success with probability p or a failure with probability 1−p. If the round is a success, then a random amount of money having an exponential distribution with rate λ is won. If the round is a failure, then the contestant loses everything that had been accumulated up to that time and cannot play any additional rounds. After a successful round, the contestant can either elect to quit playing and keep whatever has already been won or can elect to play another round. Suppose that a newly starting contestant plans on continuing to play until either the total of her winnings exceeds t or a failure occurs.

(a) What is the distribution of N, equal to the number of successful rounds that it would take until her fortune exceeds t?

(b) What is the probability the contestant will be successful in reaching a fortune of at least t?

(c) Given the contestant is successful, what is her expected winnings?

(d) What is the expected value of the contestant’s winnings?