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Chapter 5: Q. 5.22 (page 176)

Every day Jo practices her tennis serve by continually serving until she has had a total of 50successful serves. If each of her serves is, independently of previous ones,

successful with probability .4, approximately what is the probability that she will need more than 100serves to accomplish her goal?

Hint: Imagine even if Jo is successful that she continues to serve until she has served exactly 100 times. What must be true about her first 100 serves if she is to reach her goal?

Short Answer

Expert verified

The probability that Jo reaches her goal in 100serves is 0.9729.

Step by step solution

01

Step 1. Given Information.

Jo serves until she has a total of 50successful serves.

Also, probability of success is each trial is 0.4.

02

Step 2. Find mean and standard deviation of binomial distribution.

Let the random variable Xrepresent the number of serves needed.

Let n=100represent the number of trials.

Let localid="1646749828545" p=0.4represent the probability of success in each trial.

Hence, random variable Xfollows binomial distribution parameters n=100andp=0.4.

Therefore, the probability mass function of Xis expressed as

P(X=x)=nxpx1-pn-x,x=0,1,2.......n

Substitute 100forn,0.4forpin equation 1and2,

渭=np=1000.4=40蟽=np1-p=1000.40.64.898979

03

Step 3. Find the probability of serves less than or equal to 50.

PX鈮�50=鈭�x=050PX=x=鈭�x=050100x0.6100-x(0.4)x=0.0271

04

Step 4. Find the probability that Jo reaches her goal in 100 serves.

PX>100=1-PX鈮�50=1-0.0271=0.9729

Therefore, the probability that Jo reaches her goal in 100serves is0.9729.

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