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Consider a collection of $(N+1)$ urns, each containing a total of $N$ red and white balls; the urn number $K$ contains $K$ red and $N-K$ white balls $( K= 0,1,2,..., N)$. An urn is chosen at random and $n$ random drawings are made from it, the ball drawn being replaced each time. Suppose that all $n$ balls turn out to be red. Then find the probability that the next drawing will also yield a red ball.

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Sketch: Let $$p_k=P(Urn=k|all red)=P(all red|Urn= k)P(Urn= k)/P(all red).$$ Compute the probabilities on the right-hand side to evaluate $p_k$. Then the probability the next ball is red is $$\sum_{k=0}^N \frac{k}{N} p_k=\frac{1}{N}\frac{1^{n+1}+\cdots+N^{n+1}}{1^{n}+\cdots+N^{n}}.$$

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