MATH 367, fall 98, CHAPTER 5 Section 2, Oct 29 Conditional Probability

Theorem 0.1. Let A_{1}, …, A_{n} be a partition of Ω such that P (A_{i}) > 0 for all i = 1, …, n. Then,

E(X) = P (A_{1})E(X|A_{1}) + ⋯ + P (A_{n})E(X|A_{n}).

In particular, if the partition of the space is given by the values of another random variable Y, we get the following result.

Corollary 0.2.

E(X) = ∑_{y value of Y} P (Y = y)E(X|Y = y) = E(E(X|Y)).

Problems:

1. Two pennies and a nickel are tossed. The value in cents of the coins that fall heads is described by one of the words ZERO, ONE, TOW, FIVE, SIX, SEVEN. From that word we choose letters, one at a time, with replacement, until we get a vowel. Let X be the number of times a letter is chosen. Find E(X).

2. Alex and Ben play the following game: Alex writes down either number 1 or 2 and Ben must guess which one. If the number Alex has written down is i and Ben has guessed correctly, Alex pays Ben $ i. If Ben makes a wrong guess, Ben pays Alex $.75. If Ben randomizes his decision by guessing 1 with probability p and 2 with probability 1 − p, determine his expected gain.

3. What value of p maximizes Ben’s expected return? What value of p minimizes Ben’s expected return?

4. Ann buys an insurance policy against some event E from the Friendly Insurance Company. The Friendly Insurance Company agrees to pay Ann $10,000 if the event E occurs within the year the policy was bought. How much money should Ann pay for this policy if the insurance company estimates that E will occur within the year with probability p and it expects to make a 10% profit?

5. A miner is trapped in a mine containing 3 doors. The first door leads to a tunnel that will take him to safety after 3 hours of travel. The second door leads to a tunnel that will return him to the mine after 5 hours of travel. The third door leads to a tunnel that will return him to the mine after 7 hours of travel. If we assume that the miner is at all times equally likely to choose any one of the doors, what is the expected length of time until he reaches safety?

6. An urn contains 4 white balls and 6 blue balls. One ball at a time is randomly drawn until the first white ball is drawn. Find the expected number of blue balls withdrawn.

7. Suppose that the number of people entering a department store on a given day is a random variable with mean 50. Suppose further that the amounts of money spent by these customers are independent random variables with a common mean of $8. Assume also that the amount of money spent by each customer is also independent of the total number of customers to enter the store. What is the expected amount of money spent in the store on a given day?

Homework Section 5.1: read section 5.1 and do problems 2, 3, 4, 5, 6. Homework Section 5.2: read section 5.2 and do problems 9, 10, 11, 14, 15, 16.