Jane has two tennis serves, a hard serve and a soft serve.Her hard serve is in (a good serve) 50% of the time, and her soft serve is in (good) 75% of the time. If her hard

serve is in, she wins 75% of her points. If her soft serve is in, she wins 50% of her points. Since she is allowed to reserve one time if her first serve is out, what should her
serving strategy be? That is, should she serve hard followed by soft; both hard; soft followed by hard; or both soft?

Lets look at the value of each combination.

H first
H=.50*.75=.375
HH=.50*.50*.75.=.1875
HS=.50*.75*.50=.1875

S first
S=.75*.5=.375
SH=.25*.5*.75=.09375
SS=.25*.75*.50=.09375

so, check my calcs. It seems H first, then either second is the best strategy.

To determine Jane's best serving strategy, we need to consider the probabilities and outcomes for each possible sequence of serves. Let's analyze each option:

1. Serve hard followed by soft:
- The probability of her hard serve being in is 50%.
- If the hard serve is in, she wins 75% of her points.
- If the hard serve is out, she then serves soft with a probability of 75%. If the soft serve is in, she wins 50% of her points.
- Therefore, the overall probability of winning a point with this sequence is (0.5 * 0.75) + (0.5 * 0.25 * 0.5) = 0.6875.

2. Serve both hard:
- The probability of her hard serve being in is 50%.
- If both serves are in, she wins 75% of her points.
- Therefore, the overall probability of winning a point with this sequence is 0.5 * 0.75 = 0.375.

3. Serve soft followed by hard:
- The probability of her soft serve being in is 75%.
- If the soft serve is in, she wins 50% of her points.
- If the soft serve is out, she then serves hard with a probability of 50%. If the hard serve is in, she wins 75% of her points.
- Therefore, the overall probability of winning a point with this sequence is (0.75 * 0.5) + (0.25 * 0.5 * 0.75) = 0.53125.

4. Serve both soft:
- The probability of her soft serve being in is 75%.
- If both serves are in, she wins 50% of her points.
- Therefore, the overall probability of winning a point with this sequence is 0.75 * 0.5 = 0.375.

Based on the calculations above, Jane's best serving strategy is to serve soft followed by hard. This sequence has the highest overall probability of winning a point, which is 0.53125.