Sanderson M. Smith

TENNIS DILEMMA: A CUTE PROBABILITY PROBLEM

My buddy, Herkimer, has a sister named Hortense, who has potential as a tennis player. To encourage her to accept challenges, Herky offers Hortense a new car if she wins two sets in a row in a three set series to be played alternately with her mother and a tennis club champion. Hortense's mother is a good player, but not as good the tennis club champion.

Hortense has the option of choosing to play (in order):

MOTHER - CHAMPION - MOTHER

or

CHAMPION - MOTHER - CHAMPION

Which series should Hortense choose to play?

We can use MATH POWER to do a statistical analysis of this problem.

• Let c = probability Hortense beats champion in a set.
• Let m = probability Hortense beats mother in a set.
• Given information lets us conclude that m > c.

We can now calculate Hortense's probabilities of winning for each choice. Keep in mind that Hortense must win two sets in a row.

 MOTHER - CHAMPION - MOTHER PROBABILITY Hortense wins first two games mc Hortense loses first game, but wins next two games (1-m)cm

The probability Hortense wins if she chooses the order MOTHER-CHAMPION-MOTHER is

Prob(MCM) = mc + (1-m)cm = cm(1+1-m) = cm(2-m)

 CHAMPION-MOTHER-CHAMPION PROBABILITY Hortense wins first two games cm Hortense loses first game, but wins next two games (1-c)mc

The probability Hortense wins if she chooses the order CHAMPION-MOTHER-CHAMPION is

Prob(CMC) = cm + (1-c)mc = cm(1+1-c) = cm(2-c)

Now, since m > c, it follows that cm(2-m) < cm(2-c). That is, Prob(CMC) > Prob(MCM). The conclusion is that Hortense should choose the option that has her possibly having to play the champion twice.

This may seem counter-intuitive. Why would one not choose the option that provides the opportunity to play the weaker player twice? One needs to note here that Hortense must win the second match if she hopes to win two in a row under the described setup. MATH POWER demonstrates clearly that she should choose to play the "must win" match against the weaker player, which would provide her two opportunities to beat the better player.