Problem
If we assume that two completely unknown tennis players B and D are facing each other in a three set match.
- Let $p$ be the probability that B wins the first set
- Let $q$ be the probability that the match ends in two sets.
Which one of the following is true?
- $p < q$
- $p = q$
- $p > q$
My Approach
My thinking is to assume that the probability of a player winning a set is independent if the player has won a previous set and then calculate $q$ as
$q = P(BB)+P(DD)=p^2+(1-p)^2=2p^2-2p+1$
Hence, $$q=2p^2-2p+1$$
but this equation does not have real roots.
Edit
I tried the suggestion on the comments on working with a few numerical examples and constructed the following table
| $p$ | $1-p$ | $q$ |
|---|---|---|
| 0.1 | 0.9 | 0.82 |
| 0.2 | 0.8 | 0.68 |
| 0.3 | 0.7 | 0.58 |
| 0.4 | 0.6 | 0.52 |
| 0.5 | 0.5 | 0.5 |
| 0.6 | 0.4 | 0.52 |
| 0.7 | 0.3 | 0.58 |
| 0.8 | 0.2 | 0.68 |
| 0.9 | 0.1 | 0.82 |
So I guess the answer depends on the value of $p$.