Riley hobson bence · Problem 30.7

Statistics

Mathematics — Methods · probability-tournament

Problem

In a tournament of 2n2^n players, two specific players are seeded. If prp_r is the probability that before the rrth round both players are still in the tournament (and therefore have not met each other), show that pr+1=142n+1−r−22n+1−r−1prand hence thatpr=(12)r−12n+1−r−12n−1.p_{r+1} = \frac{1}{4}\frac{2^{n+1-r}-2}{2^{n+1-r}-1}p_r \quad \text{and hence that} \quad p_r = \left(\frac{1}{2}\right)^{r-1}\frac{2^{n+1-r}-1}{2^n - 1}. (a) Find the probability that they meet in the final. (b) Find the probability that they meet at some stage in the tournament.

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