Mimi just started her tennis class three weeks ago. On average, she is able to return 20% of her opponent's serves. Assume her opponent serves 8 times. Show all work. Just the answer, without supporting work, will receive no credit.

(a) Let $X$ be the number of returns that Mimi gets. As we know, the distribution of $X$ is a binomial probability distribution. What is the number of trials ($n$), probability of successes ($p$) and probability of failures ($q$), respectively?

(b) Find the probability that that she returns at least 1 of the 8 serves from her opponent.

(c) How many serves can she expect to return?

#### Solution

(a) Let $X$ be the number of returns that Mimi gets. As we know, the distribution of $X$ is a binomial probability distribution.

The number of trials $n=8$ (number of opponents serves).

The probability of success $p=0.20$ (probability that Mimi ables to return her opponent's serves.)

The probability of failure $q=0.80$ (probability that Mimi not able to return her opponent's serves.)

(b) $X\sim B(8,0.20)$. So the probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) & =\binom{8}{x} (0.20)^x (1-0.20)^{8-x}\\ &\quad x=0,1,2,\cdots, 8. \end{aligned} $$ `

The probability that she returns at least 1 of the 8 serves from her opponent is

` $$ \begin{aligned} P(X\geq 1) &= 1- P(X=0)\\ &= 1- \binom{8}{0}(0.20)^0(1-0.20)^{8-0}\\ &= 1- 0.1678\\ &= 0.8322. \end{aligned} $$ `

(c) The expected value of $X$ is `$E(X) = n\times p = 8\times 0.20 = 1.6\approx 2$`

. She expect to return 2 serves.

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators