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Exercise 1

Part of 2000 French Mathematical Olympiad

Problems(1)

maximizing probability in balls and urns

Source: French MO 2000 Exercise 1

4/9/2021
We are given bb white balls and nn black balls (b,n>0b,n>0) which are to be distributed among two urns, at least one in each. Let ss be the number of balls in the first urn, and rr the number of white ones among them. One randomly chooses an urn and randomly picks a ball from it.
(a) Compute the probability pp that the drawn ball is white. (b) If ss is fixed, for which rr is pp maximal? (c) Find the distribution of balls among the urns which maximizes pp. (d) Give a generalization for larger numbers of colors and urns.
probabilitycombinatorics