Old Jimmy and young Pete are both tennis champions. They have played each other many times, each winning exactly half of the matches. When both are fresh Jimmy is much the better player, but he tires rapidly so his probability of winning the kth set is p
k, where p is the probability that he wins the first set.
If they always play best-of-five matches, what is the value of p?
p=0.777862 +/- 0.000001
The brute force approach is to list the 10 ways Jim could win,
and set this sum of probabilities to 0.5, and then solve for p.
If Jim is A and Pete is B, the complete list of match
possibilities where Jim wins is:
AAA, BAAA, ABAA, AABA, BBAAA, BABAA, BAABA, ABBAA, ABABA,
AABBA
The ten probabilities look like:
1. p p^2 p^3 (AAA)
2. (1-p) p^2 p^3 p^4 (BAAA)
.
.
10. p p^2 (1-p^3) (1-p^4) p^5 (AABBA)
This sum is 0.5 when p is near the value above. The code is unremarkable. I found no easy way to check this result...
Edited on December 29, 2023, 11:07 pm
soln
