First slide

Introduction to probability

Question

In a knockout tournament, 2ⁿ equally skilled players; S₁, S₂, .…, $S_{2^{n}}$ are participating. In each round, players are divided in pairs at random and winner from each pair moves to the next round. If S₂ reaches the semi-final, then the probability that S₁ wins the tournament is $\frac{1}{84}$ . The value of n is ________.

Moderate

Solution

Given S₂ reaches the semi-finals.
Since all other players (2ⁿ - 1) are equally likely to win the finals with probability p.
$\begin{array}{ll} ∴ (2^{n} - 1) p + \frac{1}{4} = 1 \\ \Rightarrow (2^{n} - 1) p = \frac{3}{4} \\ \Rightarrow p = \frac{3}{4 (2^{n} - 1)} \end{array}$
If $p = \frac{1}{84}$ , then
$\begin{array}{ll} ∴ \frac{1}{84} = \frac{3}{4 (2^{n} - 1)} \\ \Rightarrow 2^{n} - 1 = 63 \\ \Rightarrow 2^{n} = 64 \\ \Rightarrow n = 6 \end{array}$

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