aez-notes

A tennis tournament has 8 players. The number a player draws from a hat decides his first-round rung in the tournament ladder. See diagram.

Suppose that the best player always defeats the next best and that the latter always defeats all the rest. The loser of the finals gets the runner-up cup. What is the chance that the second-best player wins the runner-up cup?

The diagram makes it clear that the event will occur if one draws a rung from 1–4 and the other one draws a rung from 5–8, but it does not matter which one gets the larger or smaller rung.

Without loss of generality we can consider the case where the best player draws a number from 1–4. Then there are 7 remaining rungs and 4 of them fall in 5–8. Hence the probability is 4/7.