Paper 2, Section I, F

Probability | Part IA, 2012

Given two events AA and BB with P(A)>0P(A)>0 and P(B)>0P(B)>0, define the conditional probability P(AB)P(A \mid B).

Show that

P(BA)=P(AB)P(B)P(A)P(B \mid A)=P(A \mid B) \frac{P(B)}{P(A)}

A random number NN of fair coins are tossed, and the total number of heads is denoted by HH. If P(N=n)=2nP(N=n)=2^{-n} for n=1,2,n=1,2, \ldots, find P(N=nH=1)P(N=n \mid H=1).

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