MathDB

21

Part of 1984 IMO Longlists

Problems(1)

Probability of process ending after some number of steps.

Source:

10/9/2010
(1)(1) Start with aa white balls and bb black balls. (2)(2) Draw one ball at random. (3)(3) If the ball is white, then stop. Otherwise, add two black balls and go to step 22. Let SS be the number of draws before the process terminates. For the cases a=b=1a = b = 1 and a=b=2a = b = 2 only, find an=P(S=n),bn=P(S≤n),lim⁡n→∞bna_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n, and the expectation value of the number of balls drawn: E(S)=∑n≥1nan.E(S) =\displaystyle\sum_{n\ge1} na_n.
probabilitylimitcombinatorics unsolvedcombinatorics