|
Question 209418: 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?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if her first serve is out, she gets one serve left.
if her second serve is out, she loses the point (double faults).
-----
if she serves soft she will be in 75% of the time but will only win 50% of the time after that. she will win .75 * .5 = .375 = 37.5% of the time. this means that she will lose 62.5% of the time.
-----
if she serves hard she will be in 50% of the time but will win 75% of the time after that. she will win .5 * .75 = .375 = 37.5% of the time. this means that she will lose 62.5% of the time.
-----
-----
should she do hard first and then soft or soft first and then hard or both soft or both hard?
-----
let's look at what happens:
-----
HARD FOLLOWED BY HARD
-----
she serves hard and gets it in 50% of the time.
75% of the time that she gets it in she wins the point so she'll win the point on the first serve .5 * .75 = .375 = 37.5% of the time.
-----
if she got it in 50% of the time on the first serve, then she got it out 50% of the time on the first serve so she serves hard again.
-----
she gets it in 50% of the time on the second serve and 75% of that time she wins so she wins 37.5% of the time on the second serve.
-----
however, she only serves the second time 50% of the time so she will win on the second serve .5 * 37.5% = 18.75% of the time.
-----
her total chances of winning by serving hard the first time and serving hard the second time is 37.5% on the first serve and 18.75% on the second serve.
-----
HARD FOLLOWED BY SOFT
-----
she serves hard and gets it in 50% of the time.
75% of the time that she gets it in she wins the point so she'll win the point on the first serve .5 * .75 = .375 = 37.5% of the time.
-----
if she got it in 50% of the time on the first serve, then she got it out 50% of the time on the first serve but this time she serves soft on the second serve.
-----
she gets it in 75% of the time on the second serve and 50% of that time she wins so she wins 37.5% of the time on the second serve.
-----
however, she only serves the second time 50% of the time so she will win on the second serve .5 * 37.5% = 18.75% of the time.
-----
whether she served hard or soft on the second serve didn't matter if she served hard on the first serve.
-----
SOFT FOLLOWED BY HARD
-----
she serves soft and gets it in 75% of the time.
50% of the time that she gets it in she wins the point so she'll win the point on the first serve .75 * .5 = .375 = 37.5% of the time.
-----
if she got it in 75% of the time on the first serve, then she got it out 25% of the time on the first serve so she serves the second time hard.
-----
she gets it in 50% of the time on the second serve and 75% of that time she wins so she wins 37.5% of the time on the second serve.
-----
however, she only serves the second time 25% of the time so she will win on the second serve .25 * 37.5% = 9.375% of the time.
-----
her total chances of winning by serving soft the first time and serving hard the second time is 37.5% on the first serve and 9.375% on the second serve.
-----
SOFT FOLLOWED BY SOFT
-----
she serves soft and gets it in 75% of the time.
50% of the time that she gets it in she wins the point so she'll win the point on the first serve .75 * .5 = .375 = 37.5% of the time.
-----
if she got it in 75% of the time on the first serve, then she got it out 25% of the time on the first serve so she serves the second time soft again.
-----
she gets it in 75% of the time on the second serve and 50% of that time she wins so she wins 37.5% of the time on the second serve.
-----
however, she only serves the second time 25% of the time so she will win on the second serve .25 * 37.5% = 9.375% of the time.
-----
whether she served hard or soft on the second serve didn't matter if she served soft on the first serve.
-----
however, ...
if she served hard on the first serve, then she won 37.5% of the time on the first serve and 18.75% on the second serve.
if she served soft on the first serve, then she won 37.5% of the time on the first serve and only 9.375% on the second serve.
-----
it appears she should serve hard first and it doesn't really matter if she serves hard or soft on the second serve.
-----
this kind of makes sense.
her chances of winning on the first serve or the second serve by themselves are equal whether she serves hard or soft.
-----
if she serves hard first, then she gets a chance to win on the second serve 50% of the time.
if she serves soft first, then she gets a chance to win on the second serve only 25% of the time.
-----
her probability of winning on the first serve is the same but her probability of winning on the second serve is greater even though the probability of winning on the second serve is the same because she gets more chances on the second serve if she serves hard first.
-----
with the same probability of getting a hit each time, your chances of getting a hit are greater the more times you come to bat.
-----
serve hard first looks to be the way to go. doesn't really matter what you do after that.
-----
|
|
|
| |
