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In a knockout tournament 2^n equally ski...

In a knockout tournament `2^n` equally skilled players, `S_1,S_2, S_(2n)` are participating. In each round, players are divided in pair at random and winner form each pair moves in the next round. If `S_2` reaches the semi-final, then the probability that `S_1` wins the tournament is 1/84. The value of `n` equals__________.

Text Solution

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Given `S_(2)` reaches the semi-finals.
Since all other players `(2^(n) - 1)` are equally likely to win the finals with probability p.
`therefore (2^(n)-1)p+(1)/(4) = 1`
`(2^(n) - 1)p = (3)/(4)`
implies `p=(3)/(4(2^(n) - 1))`
If `p = (1)/(84)`, then
`therefore(1)/(84) = (3)/(4(2^(n) - 1))`
`implies 2^(n) - 1 = 63`
`implies 2^(n) = 64`
implies n = 6
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