First slide

Introduction to probability

Question

Forty teams play a tournament. Each team plays every other team just once. Each game results in a win for one team. If each team has a 50% chance of winning each game, the probability that at the end of the tournament, every team has won a different number of games is

Moderate

A

$\frac{1}{780}$
B

$\frac{40!}{2^{780}}$
C

$\frac{40}{2^{780}}$
D

none of these

Solution

Team totals must be 0, 1, 2, ..., 39. Let the teams be T₁, T₂, ..., T₄₀, so that T_i loses to T_j for i < j.
In other words, this order uniquely determines the result of every game.
There are 40! such orders and 780 games, so 2⁷⁸⁰ possible outcomes for the games.
Hence, the probability is 40!/2⁷⁸⁰.

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