Paper 2, Section II, F

Let $\Omega$ be the sample space of a probabilistic experiment, and suppose that the sets $B_{1}, B_{2}, \ldots, B_{k}$ are a partition of $\Omega$ into events of positive probability. Show that

$\mathbb{P}\left(B_{i} \mid A\right)=\frac{\mathbb{P}\left(A \mid B_{i}\right) \mathbb{P}\left(B_{i}\right)}{\sum_{j=1}^{k} \mathbb{P}\left(A \mid B_{j}\right) \mathbb{P}\left(B_{j}\right)}$

for any event $A$ of positive probability.

A drawer contains two coins. One is an unbiased coin, which when tossed, is equally likely to turn up heads or tails. The other is a biased coin, which will turn up heads with probability $p$ and tails with probability $1-p$. One coin is selected (uniformly) at random from the drawer. Two experiments are performed:

(a) The selected coin is tossed $n$ times. Given that the coin turns up heads $k$ times and tails $n-k$ times, what is the probability that the coin is biased?

(b) The selected coin is tossed repeatedly until it turns up heads $k$ times. Given that the coin is tossed $n$ times in total, what is the probability that the coin is biased?

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