The Table-Tennis Tournament

Probability Level 4

I show up to a table-tennis tournament with 9 9 people. I decide to join. Then the tournament starts. The tournament is round robin (you play each other once). The probability that I win any table-tennis game is 1 4 \frac{1}{4} . The probability that after the tournament is over I won at least 1 1 game can be written as m n \frac{m}{n} . Find m + n m+n


The answer is 504605.

Alex Segesta by Alex Segesta , 20, USA

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1 solution

Since you are going to play once against every other player, you are going to play 9 games. The probability that you win a game is 1 4 \frac{1}{4} , hence the probability that you lose a game is 1 − 1 4 = 3 4 . 1-\frac{1}{4}=\frac{3}{4}. To win at least one game during the tournament, the only possible situation that can't happen is you lose all games. The probability that you lose all games is given by P = ( 3 4 ) 9 P=(\frac{3}{4})^{9} , hence our final answer is: P ′ = 1 − P = 1 − ( 3 4 ) 9 = 242461 262144 P' = 1-P = 1 - (\frac{3}{4})^{9} = \frac{242461}{262144} . m n = 242461 262144 ⟶ m = 242461 , n = 262144 \frac{m}{n} = \frac{242461}{262144} \longrightarrow m=242461 \, , n=262144 m + n = 504605 \large \boxed{m+n=504605}

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