Q. 5.22 - A First Course in Probability

Question

Every day Jo practices her tennis serve by continually serving until she has had a total of $50$ successful 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 $100$ serves 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?

Step-by-Step Solution

Answer

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

Step 1. Given Information.

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

Also, probability of success is each trial is $0.4$.

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

Let the random variable $X$ represent the number of serves needed.

Let $n=100$ represent the number of trials.

Let $p=0.4$ represent the probability of success in each trial.

Hence, random variable $X$ follows binomial distribution parameters $n=100 and p=0.4$.

Therefore, the probability mass function of $X$ is expressed as

$P(X=x) = \left[\frac{n}{x}\right]p^x{\left(1-p\right)}^{n-x}, x=0,1,2.......n$

Substitute $100 for n, 0.4 for p$in equation $\left(1\right) and \left(2\right)$,

$$\begin{gathered} \mu =np \\ =100\left(0.4\right) \\ =40 \\ \sigma =\sqrt{np\left(1-p\right)} \\ =\sqrt{100\left(0.4\right)\left(0.6\right)} \\ 4.898979 \end{gathered}$$

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

$$\begin{gathered} P\left(X\le 50\right)=\sum _{x=0}^{50}P\left(X=x\right) \\ = \sum _{x=0}^{50}\left(\begin{array}{c}100\\ x\end{array}\right){\left(0.6\right)}^{100-x}{(0.4)}^x \\ =0.0271 \end{gathered}$$

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

$$\begin{gathered} P\left(X>100\right)=1-P\left(X\le 50\right) \\ =1-0.0271 \\ =0.9729 \end{gathered}$$

Therefore, the probability that Jo reaches her goal in $100$ serves is $0.9729$.