Managing Risk

MaManaging Risk and Uncertainty: Exercises 1 1 List or describe Ω, the set of outcomes for each of the following experiments: a) tossing a coin twice; b) the score in a soccer match; c) the time to wait until the next pulsar is discovered; d) the amount by which the annual income of a woman exceeds that of her husband.

2 Suppose that Ω = {1, 2, 3, . . .} with pω = 1/(ω(ω + 1)) for ω ≥ 1. Verify that this is a genuine probability space. Find the probabilities of the following events: a) getting at least 5; b) getting an odd number.

3 In the UK National Lottery, where six numbers are chosen at random from a list of 49 numbers, players select six numbers themselves, hoping to match as many of the chosen numbers as possible. Find the probabilities that for a given entry: a) exactly four winning numbers are selected; b) at least four winning numbers are selected; c) exactly two of the winning numbers are multiples of six.

4 The roulette wheel in a UK casino has 37 numbers, {0, 1, . . . , 36}, all equally likely. Pandora makes bets on the following combinations: a) top half = {19, 20, 21, . . . , 36}; b) odd = {1, 3, 5, . . . , 35}; c) bottom row = {34, 35, 36}; d) the foursome {14, 15, 17, 18}.

Her bet wins if the winning number falls into her selection. Let A, B, C and D indicate that the above four respective bets are winning bets. Find the probabilities of A, B, A∩B, A ∪ B, B ∩ C and A ∩ B ∩ C ∩ D.

1

5 Ninety percent of children are vaccinated. Five per cent of unvaccinated children will catch measles, as will one in 200 of vaccinated children. a) What proportion of children will catch measles? b) Find the probability that someone had been vaccinated, conditional on them catching measles.

6 An urn initially has one red ball. Persephone uses a device to select n blue balls with probability e−λ λn /n!, for n = 0, 1, 2, . . ., and add them to the urn. She then selects one ball at random from the urn. Show that the probability that she selects the...