1. Suppose a fair die is rolled twice, independently. Find the probabilities of each of the following pairs of events. Which pairs of events are independent? Explain your answers.

(a) The fifirst roll is a two; the second roll is odd.[3 marks]

(b) The fifirst roll is a two; the two rolls are equal.[3 marks]

(c) The fifirst roll is a two; the sum of both rolls is six.[3 marks]

1. A bowl contains 20 cherries, exactly 15 of which have had their pits removed. A greedy pig eats fifive whole cherries, picked uniformly at random, without remarking on the presence or absence of pits. Subsequently, a cherry is picked uniformly at random from the remaining fififteen.

(a) What is the probability that this cherry contains a pit?[2 marks]

(b) Given that this cherry contains a pit, what is the probability that the pig consumed at least one cherry pit?[4 marks]

1. Let (⌦, F, P) be a probability space and let A 2 F with P(A) > Defifine the probability measure P0 on (⌦, F) by P0(B) = P (B | A)8B 2 F. Prove that P0 is a probability measure, justifying all your steps.[5 marks]
1. NASA is developing two top-secret space shuttles. One has two engines, the other has four. All the engines are identical, and have the same probability of failure. Each shuttle is designed to flfly if at least half of its engines work. A visiting scientist says, “The fourengine shuttle is more reliable, isn’t it?” The NASA technician replies that the probability of failure is top-secret, but that in fact both shuttles have the same probability of flflying.

The visitor then says, “Aha! Never mind, now I know both the probability an engine will fail and the probability that the shuttle will flfly.” How did the visitor fifigure this out, and what are the two probabilities?[5 marks]

1. Let (⌦, F) be a measurable space. Defifine the “conditioning” functionc(P, A) = P (· | A)for P a probability measure on (⌦, F) and A an event with P(A) > 0.

That is, write M1(⌦, F) for the set of probability measures on (⌦, F). Then c is a function from {(P, A)2 M1(⌦, F)F : P(A) > 0} to M1(⌦, F), where c(P, A) is the probability measure P0 defifined by P0(B) =P(B | A) for any B 2 F. In Question 3,c(P, A) was simply called P0 , but this question applies conditioning several times and uses a more  elaborate notation.

(a) Show that c(c(P, A), B) = c(P, A \ B) for all A, B 2 F such that P(A \ B) > 0.

Hint: unravel what this statement means. It may be helpful to give names like P0, P00  to some of the probability measures obtained from c. [4 marks]

(b) In words or symbols, interpret the result of part (a) without reference to the function c.[1 mark]

1. * Let D and T be sets, and let X : D ! T be a function from D to T. Let T be a σ-algebra of subsets of T, and defifine D =! X1(B): B 2 T, the collection of subsets A D of the form A = X1(B) forB 2 T .

(a) * Show directly that D is a σ-algebra of subsets of D.[3 marks]

Now let T = R andT = B, the Borel σ-algebra. Let ⌦= D, let F be a σ-algebra of subsets of ⌦, and let P be a probability measure on (⌦, F).

(a) * Suppose that X is a random variable on the probability space (⌦, F, P). Showthat D ⇢ F.[2 marks] EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

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