Exercises

Introductory questions

These are questions that you might already be able to do! They cover some of the main ideas we’ll need in the course: some of them can be answered systematically, and others emphasise the importance of clearly defining the space we’re working in.

Warm-up

Exercise 1 In a film dramatizing the early history of probability, three actors are required to portray three mathematicians, each with a distinctive (false) beard. Due to a mix-up in the make-up department, the false beards are randomly distributed to the actors (one beard each). What is the probability that at least one beard goes to the right actor?

Exercise 2  

1. On the last day of a basketball season, only teams A and B can possibly win the league. Team A will win the league if either they win their last game, or if Team B lose their last game (A and B are not playing each other). Games do not end in draws, and we naively assume that all outcomes are equally likely. What is the probability of A winning the league?

2. Now suppose it is a football season, and draws are possible. Again we assume all outcomes are equally likely, and that A needs to do at least as well as B to in the league. What is the probability of A winning the league?

Exercise 3 Three dice are rolled. Find the probability that the total score is (a) 3; (b) 4; (c) 5; (d) 6.

Exercise 4 Three shots are fired at a target that is divided into three regions. Each shot hits the target and is equally likely to hit any of the three regions. Find (a) the probability that all three regions are hit; (b) the probability that precisely two regions are hit.

Exercise 5 Two chess players want to decide fairly who should play white in a particular game by flipping a coin. Unfortunately, the only coin that they have is biased to heads. How can they still decide fairly by flipping?

Workout

Exercise 6 The good, the bad, and the uncertain: A, B and C are to fight in a three-cornered pistol duel (a `truel’). All know that A’s chance of hitting his target is 0.3, C’s is 0.5, and B never misses. They are to fire at their choice of target one after the other, in the order A, B, C, cyclically (with gunmen who are hit taking no further part), until only one man remains. What is A’s best strategy?

Exercise 7 Mr and Mrs Smith have two children.

1. They call one of them, Derek (a boy), down from his room and introduce you. What is the probability that both of the Smiths’s children are boys?
2. They tell you their eldest child is Derek (a boy). Does this change the probability that both of the Smiths’s children are boys? If so, how?

Assume that boys and girls are equally likely.

Exercise 8 Jimmy the gambler has three dice, each with an unusual configuration of spots on their six faces: \[\begin{align*} & \text{Die A has faces:} \quad 1, 1, 4, 4, 4, 4 ;\\ & \text{Die B has faces:} \quad 3, 3, 3, 3, 3, 3 ;\\ & \text{Die C has faces:} \quad 2, 2, 2, 2, 5, 5 . \end{align*}\] Jimmy picks a die, I pick a die, and we roll for high stakes, the larger number winning. Jimmy lets me pick the first die. Is he being nice? Which die should I pick?

Stretch

Exercise 9  

1. A recent study showed that most great male mathematicians were eldest sons (great female mathematicians were not studied). Must this mean that first-born sons are more likely to have mathematical ability than sons born later?
2. Another study showed that a far greater proportion of the population died of tuberculosis (TB) in Arizona than in any other state in America. Must this mean that Arizona’s climate favours getting TB?
3. Yet another study was carried out on alleged sex-discrimination in graduate school admissions at Berkeley, California. About 44% of men applying for graduate work were accepted, whereas only 35% of women applying for graduate work were accepted. The qualifications of men and women were roughly the same. Surprisingly, when the data were examined to identify the departments where discrimination occurred, it turned out that in each department women had a greater chance of being accepted than men. How can this apparent paradox be explained?