An auto manufacturer has sold 50,000 vehicles with a defective gonculator.As required by law, the manufacturer has issued a recall notice and will replace the part at no cost to the customer.

Based on similar recall notices in the past, the firm believes that for each of the 50,000 vehicle owners, there is a 30% chance that the owner will ignore the recall notice, and a 70% chance he/she will respond.

If a customer does not respond to the recall notice, the firm incurs no cost ($0).For any givencustomer that responds, there are two possibilities for the manufacturer:

• with 60% probability the responding customer's gonculator can simply be repaired at a cost of $50;
• with 40% probability the responding customer's gonculator must be replaced at a cost of $150.

Therefore notice that there are three possible cost outcomes for each of the 50,000 customers.

(a) Let X denote the recall cost incurred by the manufacturer for any one randomly chosen vehicle customer. Give the probability distribution of X (for example, via a table).

(b) What is theexpectedtotal costfrom this recall notice that the firm incurs across all 50,000 customers?

(c) Assume that the cost incurred for any one customer is independent of that for any other customer. Then there is a 90% chance that thetotal costincurred from this recall notice will be $_________ or less.

(d) Assume that the cost incurred for any one customer is independent of that for any other customer, and suppose that the manufacturer sets aside $3.16 million to cover the costs of the recall notice. What is the probability that they do not cover all the costs?