aez-notes
Successive Wins
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 alternatively; father-champion-father or champion-father-champion, according to Elmer's choice. The champion is a better player than Elmer's father. Which series should Elmer choose?
The possible outcomes of the game series are shown in the following table.
| Game 1 | Game 2 | Game 3 | Wins series |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
Since the champion is better than father, we assume that the probability of the beating the champion, \(c\), is less than that of beating father, \(f\). The number of outcomes is small enough that we can write this out explicitly and just ask maxima to check if the statement is true.
assume(f > c, c > 0); cfc_win_prob: (1-c)*f*c + c*f*(1-c) + c*f*c; fcf_win_prob: (1-f)*c*f + f*c*(1-f) + f*c*f; print(is(cfc_win_prob > fcf_win_prob));
This prints out "true", telling us that the better series for Elmer to choose is CFC.