### Pollard homework sheet 4

#### Due: Thursday 8 October

(4.1)
A contestant, Fred, on a television show gets to choose one of three boxes A, B, or C. One box contains a big prize of \$10,000; the other two boxes are empty. Fred chooses box A. The host offers Fred \$500 if he will surrender the box and give up the chance for the big prize. Fred refuses. The host then opens box B, revealing it to be empty, and offers Fred \$1000 to surrender his box. Fred again refuses, but asks to switch his choice to box C.

Assume that the prize is placed in a box at random, and that Fred makes his choice at random. Assume also that the host knows where the prize is, and that he would not have revealed its location by accident.

Calculate the conditional probability that box A contains the prize, given what Fred learns from the host. Explain carefully why Fred might act as he does.

(4.2)
I have two coins in my pocket. One of them is fair (probability 1/2 of landing heads on any toss) and the other is weighted to land heads with probability 2/3. I choose a coin at random and toss it twice. Write H1 for the event that the first toss lands heads and H2 for the event that the second toss lands heads.

1. List the elments in a suitable sample space for this experiment.
2. Assign probabilities to each outcome in your sample space.
3. Find P(H1) and P(H2).
4. Find P(H2 | H1)
5. Are the events H1 and H2 independent? Explain.

(4.3)
The kingdom being aflicted with a surfeit of males, the ruler declares a new reproductive policy:
(a) no couple may have more than four children;
(b) after the birth of a male child, a couple may produce no more children.

Assume that each new birth has probability 0.5 of producing a male, regardless of previous reproductive history, and that all couples continue to produce children until either they have a son or they reach the maximum number of 4 children.

Write X for the number of female offspring in a family and Y for the number of male offspring.

1. Construct a sample space, complete with probabilities assigned to each outcome, to describe the reproductive "experiment" for a couple.
2. Find the distribution of X and the distribution of Y.
3. Find the mean of X and the mean of Y.
4. Find P{X > Y} and P{X < Y}.
5. Will the ruler's new policy succeed in reducing the proportion of males in the kingdom? Explain.

Each Section leader will assign extra problems. Some Section leaders might choose to have you solve not all of the Pollard problems.