# Tennis in 2 or 3 sets

In a best of 3 tennis match, the player that first wins 2 sets wins the match. For a 3-set tennis match, would you bet on it finishing in 2 or 3 sets?

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#### Solution

2 sets, regardless of how likely each player is to win.

Mathematical Solution

Let’s say the probability of player 1 winning is p.

To finish in 2 sets:

• Either player 1 wins the first 2 sets = p^2
• Or player 2 wins the first 2 sets = (1 – p)^2
• So P(2 sets) = p^2 + (1 – p)^2

To finish in 3 sets:

• Either player 1 wins the first set and player 2 wins the second set = p * (1 – p)
• Or player 2 wins the first set and player 1 wins the second set = (1 – p) * p
• So P(3 sets) = 2 * p * (1 – p)

P(2 sets) – P(3 sets) = p^2 + (1 – p)^2 – 2 * p * (1 – p)

This can be factored into a perfect square (p – (1 – p))^2 or just simplified to 4p^2 + 4p + 1. Either way, it’s clear that P(2 sets) – P(3 sets) ≥ 0, so it’s always better to bet on the match finishing in 2 sets.

Intuitive solution

We just need to figure out if it is more likely the same player wins the first 2 sets or different players win the first 2 sets. It’s more likely that the player that wins the first set is the better player, and therefore it’s more likely that the same (better) player wins the second set, so you should bet on the match finishing in 2 sets.