Question 1. Historically a bank expects about 5% of its borrowers to default (not repay). The bank currently has 250 loans outstanding.

(a) In order to use a binomial model to compute the probabilities associated with defaults, what must the bank assume about the behavior of these borrowers?

(b) Do the necessary assumptions listed in part (a) appear reasonable in the context of this problem?

(c) The bank has reserves on hand to cover losses if 25 of these loans were to default. Will these reserves will be enough?

Question 2. A dairy farmer accidentally allowed some of his cows to graze in a pasture containing weeds that would contaminate the milk from this herd. The farmer estimates that there’s a 10% chance of a cow grazing on some of the flavorful weeds.

(a) Under these conditions, what is the probability that none of the 12 animals in this herd ate the tasty weeds?

(b) Does the Poisson model give a good estimate of the probability that no animal ate the weed?

Question 3. A tire manufacturer warranties its tires to last at least 20,000 miles or “you get a new set of tires.” In its experience, a set of these tires lasts on average 26,000 miles with a standard deviation of 5,000 miles. Assume that the wear is normally distributed. The manufacturer profits $200 on each set sold, and replacing a set costs the manufacturer $400.

(a) What is the probability that a set of tires wears out before 20,000 miles?

(b) What is the probability that the manufacturer turns a profit on selling a set to one customer?

(c) If the manufacturer sells 500 sets of tires, what is the probability that it earns a profit after paying for any replacements? Assume that the purchases are made around the country and that the drivers experience independent amounts of wear.