Probability | Part IA, 2002

(a) Define the conditional probability P(AB)P(A \mid B) of the event AA given the event BB. Let {Bi:1in}\left\{B_{i}: 1 \leq i \leq n\right\} be a partition of the sample space Ω\Omega such that P(Bi)>0P\left(B_{i}\right)>0 for all ii. Show that, if P(A)>0P(A)>0,

P(BiA)=P(ABi)P(Bi)jP(ABj)P(Bj).P\left(B_{i} \mid A\right)=\frac{P\left(A \mid B_{i}\right) P\left(B_{i}\right)}{\sum_{j} P\left(A \mid B_{j}\right) P\left(B_{j}\right)} .

(b) There are nn urns, the rr th of which contains r1r-1 red balls and nrn-r blue balls. You pick an urn (uniformly) at random and remove 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 that i=1n1i(i1)=13n(n1)(n2)\sum_{i=1}^{n-1} i(i-1)=\frac{1}{3} n(n-1)(n-2).]

(c) What is meant by saying that two events AA and BB are independent?

(d) Two fair dice are rolled. Let AsA_{s} be the event that the sum of the numbers shown is ss, and let BiB_{i} be the event that the first die shows ii. For what values of ss and ii are the two events As,BiA_{s}, B_{i} independent?

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