V. Bayes' Rule

Suppose that 20% of drivers in Hawaii get into an automobile accident every year. Of this group, 50% will also have gotten a speeding ticket that same year. By contrast, only 10% of those who do not get into an automobile accident will have gotten a speeding ticket that year. Given that a driver selected at random has recently gotten a speeding ticket, what is the probability that they had (or will have) gotten into an automobile accident this year?

On the popular TV show, 24 Hours to Hell and Back, Master Chef Gordon Ramsay and his team try to turn around failing restaurants by giving advice to the owner(s) about how to improve their business. He then revisits the restaurants one year later to see if they are still in business and follows up with the staff. In an episode last year, he finds that among the restaurants that are still open, 75% followed his advice, while among the restaurants that are not still open, only 50% took his advice. Suppose further that most of the restaurants were already struggling financially, so assume that only about 40% of them would have been able to survive another year.

Use Bayes' rule to determine the probability that the restaurant that Ramsay visits remains open the following year given that the owner(s) of the business took his advice.

What is the probability that this restaurant will still be open the year in the year to come given that they continue to follow Ramsay's advice?

VI. Expected Value

Imagine a game where one selects a single bill at random from a bag containing the following quantities and denominations of currency: ten $1 bills, five $2 bills, three $5 bills, one $10 bill, and one $100 bill. The player gets to keep the selected bill. There is a $10 entry fee to play.

a. What is the expected value of the game?

b. What should the entry fee be to make it a "fair" game?

An auto insurance company charges $150 for a collision policy that will pay for one accident a year. For a major accident, the policy pays $5000; for a minor accident, the policy pays $1000. The $150 premium is non-refundable if the driver does not get into an accident during the year. The company estimates that the probability of a major accident is 0.005 and that the probability of a minor accident is 0.08.

a. Construct a decision table with the relevant information. Label the rows with the possible actions: "major accident," "minor accident," and "no accident." Label the columns with the possible outcomes: "major accident payout," "minor accident payout," and "no payout." Fill it in based on the information presented in the passage.

b. What is the expected value of the collision policy for the insurance company?

c. If the company sells 100,000 such policies this year, how much can it expect to earn?

Hawaiian Airlines is considering adding a flight route to the city of Austin, Texas given the rapid growth of the city as the biotech industry shifts its operations to more affordable parts of the country. Market research indicates that if Hawaiian Airlines opens up the additional flight route, it has a 42% probability of making an additional $30 million profit a year, a 22% probability of breaking even, and a 36% probability of losing $40 million.

a. What is the expected value monetary of adding a route to Austin?

b. What other considerations might motivate Hawaiian Airlines to add the route?