IMO 1984 LL FRG21

(1) Start with a white balls and b black balls.

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IMO 1984 LL FRG21

Origin: FRG

Problem

(1) Start with a white balls and b black balls. (2) Draw one ball at random. (3) If the ball is white, then stop. Otherwise, add two black balls and go to step 2. Let S be the number of draws before the process terminates. For the cases a = b = 1 and a = b = 2 only, find an = P(S = n), bn = P(S \leq n), limn\to\inftybn, and the expectation value of the number of balls drawn: E(S) =  n\geq1 nan.