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Meeting in a bracket

ProbabilityEasy

A knockout tournament has $16$ players placed in the bracket uniformly at random. Every match eliminates the loser. Anna and Bohdan are $2$ of the players. What is the probability that they play each other at some point? Give a fraction a/b in lowest terms.

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Hint

Count how many matches are played in total and how many distinct pairs of players exist.

Solution

Each match eliminates exactly $1$ player, so a tournament with $16$ players has $15$ matches.
Every match is played by a distinct pair of players, and by symmetry every pair is equally likely to be one of them.
The number of pairs of players is $(2 16) = 120$ .
So the probability that Anna and Bohdan meet is $\frac{15}{120} = \frac{1}{8}$ .
The answer is $\frac{1}{8}$ .

Answer: $\frac{1}{8}$

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