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?

both hard

To determine the optimal serving strategy for Jane, we need to consider the probability of each serve going in and the corresponding probability of winning a point with each serve.

Let's calculate the overall probability of winning a point for each serving strategy:

1. Serve Hard followed by Soft (HS):
- The probability of the hard serve going in is 50% (0.5).
- If the hard serve is in, Jane wins 75% (0.75) of her points.
- If the hard serve is out, she will serve soft next.
- The probability of the soft serve going in is 75% (0.75).
- If the soft serve is in, Jane wins 50% (0.5) of her points.

So the overall probability of winning a point with this strategy is:
0.5 * 0.75 + 0.5 * 0.75 * 0.5 = 0.75

2. Serve Hard twice (HH):
- The probability of the hard serve going in is 50% (0.5).
- If the hard serve is in, Jane wins 75% (0.75) of her points.
- Since she doesn't have a second serve, she will lose the point if her first serve is out.

So the overall probability of winning a point with this strategy is:
0.5 * 0.75 = 0.375

3. Serve Soft followed by Hard (SH):
- The probability of the soft serve going in is 75% (0.75).
- If the soft serve is in, Jane wins 50% (0.5) of her points.
- If the soft serve is out, she will serve hard next.
- The probability of the hard serve going in is 50% (0.5).
- If the hard serve is in, Jane wins 75% (0.75) of her points.

So the overall probability of winning a point with this strategy is:
0.75 * 0.5 + 0.25 * 0.75 * 0.75 = 0.53125

4. Serve Soft twice (SS):
- The probability of the soft serve going in is 75% (0.75).
- If the soft serve is in, Jane wins 50% (0.5) of her points.
- Since she doesn't have a second serve, she will lose the point if her first serve is out.

So the overall probability of winning a point with this strategy is:
0.75 * 0.5 = 0.375

Based on the calculations, the strategy that gives Jane the highest overall probability of winning a point is to serve Soft followed by Hard (SH) with an overall probability of 0.53125.

Therefore, Jane should serve Soft followed by Hard as her serving strategy.