You are allowed to hit three serves. Object of the game is to see how fast your fastest can be (it has to go in). Assume there is a linear relationship between serve speed and %chance of it going on (for simplicity sake lets say that you have a 100% chance of making a 50mph serve and a 0% chance of making a 120mph serve, and it is linear between those 2 speeds). If you miss all 3 serves, your score is 0. Otherwise, your score is the same as your fastest serve that went in (in mph).

SO....

How do you maximize your expected score? Do you start off with an easy serve, then go for pregressively faster serves? Or perhaps go for a medium speed serve first?

Is it possible to fit an equation to this problem that would tell you how fast a serve you should go for on the first, 2nd, and 3rd try (obviously the 2nd and 3rd serves would depend on whether you made the serves before it)? To make the problem more difficult (for those that are really math buffs), how would the equation change if you were given 4 serves or 5 or 10? Also, how would the equation change if we recognize that it is not a linear relationship between speed and chances of going in?

Discuss.

Best,
-Grant