In a knockout tournament, $$2n$$ equally skilled players; $$S1$$, $$S2$$, .…, $$S2n$$ 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 $$184$$. The value of n is $$________$$.

Question

In a knockout tournament, $$2n$$ equally skilled players; $$S1$$, $$S2$$, .…, $$S2n$$ 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 $$184$$. The value of n is $$________$$.

Infinity Learn · Accepted Answer

Given $$S2$$ reaches the $$semi-finals$$.Since all other players (2n$$ $$-$$ $$1) are equally likely to win the finals with probability p.∴     $$2n$$−$$1p+14=1$$⇒     $$2n$$−$$1p=34$$⇒ $$p=342n$$−$$1If$$ $$p=184$$, then∴    $$184=342n$$−$$1$$⇒    $$2n$$−$$1=63$$⇒     $$2n=64$$⇒     $$n=6$$

Introduction to probability

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Introduction to probability

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Given S2 reaches the semi-finals.Since all other players (2n - 1) are equally likely to win the finals with probability p.∴ 2n−1p+14=1⇒ 2n−1p=34⇒ p=342n−1If p=184, then∴ 184=342n−1⇒ 2n−1=63⇒ 2n=64⇒ n=6

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