Find the average premium received per policyholder of the insurance company of Example 4.27 if λ = 1/4 for one-third of its clients, and λ = 1/2 for two-thirds of its clients. Suppose that on each...

Find the average premium received per policyholder of the insurance company of Example 4.27 if λ = 1/4 for one-third of its clients, and λ = 1/2 for two-thirds of its clients. Suppose that on each play of the game a gambler either wins 1 with probability p or loses 1 with probability 1 − p. The gambler continues betting until she or he is either up n or down m. What is the probability that the gambler quits a winner?