Paper 2, Section II, 9E

Probability | Part IA, 2021

(a) (i) Define the conditional probability P(AB)\mathbb{P}(A \mid B) of the event AA given the event BB. Let {Bj:1jn}\left\{B_{j}: 1 \leqslant j \leqslant n\right\} be a partition of the sample space such that P(Bj)>0\mathbb{P}\left(B_{j}\right)>0 for all jj. Show that, if P(A)>0\mathbb{P}(A)>0,

P(BjA)=P(ABj)P(Bj)k=1nP(ABk)P(Bk)\mathbb{P}\left(B_{j} \mid A\right)=\frac{\mathbb{P}\left(A \mid B_{j}\right) \mathbb{P}\left(B_{j}\right)}{\sum_{k=1}^{n} \mathbb{P}\left(A \mid B_{k}\right) \mathbb{P}\left(B_{k}\right)}

(ii) There are nn urns, the rr th of which contains r1r-1 red balls and nrn-r blue balls. Alice picks an urn (uniformly) at random and removes two balls without replacement. Find the probability that the first ball is blue, and the conditional probability that the second ball is blue, given that the first is blue. [You may assume, if you wish, that i=1n1i(i1)=13n(n1)(n2)\sum_{i=1}^{n-1} i(i-1)=\frac{1}{3} n(n-1)(n-2).]

(b) (i) What is meant by saying that two events AA and BB are independent? Two fair (6-sided) dice are rolled. Let AtA_{t} be the event that the sum of the numbers shown is tt, and let BiB_{i} be the event that the first die shows ii. For what values of tt and ii are the two events AtA_{t} and BiB_{i} independent?

(ii) The casino at Monte Corona features the following game: three coins each show heads with probability 3/53 / 5 and tails otherwise. The first counts 10 points for a head and 2 for a tail; the second counts 4 points for both a head and a tail; and the third counts 3 points for a head and 20 for a tail. You and your opponent each choose a coin. You cannot both choose the same coin. Each of you tosses your coin once and the person with the larger score wins the jackpot. Would you prefer to be the first or the second to choose a coin?

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