Successive Wins, from Fifty Challenging Problems in Probability

November 13, 2023

The author poses the following problem:

To encourage Elmer’s promising tennis career, his father offers him a prize if he wins (at least) two tennis sets in a row in a three-set series to be played with his father and the club champion alternately: father-champion-father (FCF) or champion-father-champion (CFC), according to Elmer’s choice. The champion is a better player than Elmer’s father. Which series should Elmer choose?

The answer is CFC, and the author emphasizes, after listing all the possible sequences, the importance of the middle match where a victory must absolutely be scored. The easier the middle match is, the better.

This explanation did not resonate much with me. Neither did the algebra solution I came up with:

P(W | CFC) = P(C) * P(F) + (1 - P(C)) * P(F) * P(C)
           = (P(C) * P(F)) * (2 - P(C)) 

P(W | FCF) = P(F) * P(C) + (1 - P(F)) * P(C) * P(F) 
           = (P(C) * P(F)) * (2 - P(F))

Where P(C) is the Probability of winning against the champion and P(F) that of winning against the father.

The logic for the algebra is that there are two ways of winning: Win the first two matches, or lose the first match and then win the next two. Also P(C) * P(F) = P(F) * P(C)

Reading the last lines, since P(C) < P(F), we have 2 - P(C) > 2 - P(F), and thus P(W | CFC) > P(W | FCF)

This solution doesn’t give me any intuition. It suggests the CFC advantage comes from a higher probability of losing the first match 1 - P(C) vs 1 - P(F) in FCF. Why would a higher probability of losing one match be better?

I found another way to make sense of the solution. Thinking in extremes, what if there were zero chances of losing against the father, meaning P(F) = 1? Then:

P(W | CFC) = P(C) + (1 - P(C)) * P(C)
P(W | FCF) = P(C)

This immediately suggests that the advantage of CFC comes from having two chances at beating the strongest opponent. In this scenario, this is always better than having two changes at beating the weakest opponent.

Obviously, to get two chances at beating the champion, Elmer has to face the father in the second match, as the author emphasizes. Sounds like a half-full vs half-empty glass situation. One is just much more obvious for me.