A particular telephone number is used to receive both voice calls and fax messages. Suppose that 20% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls. (Round your answers to three decimal places.)

(a) What is the probability that at most 8 of the calls involve a fax message?

(b) What is the probability that exactly 8 of the calls involve a fax message?

(c) What is the probability that at least 8 of the calls involve a fax message?

(d) What is the probability that more than 8 of the calls involve a fax message?

#### Solution

Here $X$ denote the number of incoming calls involve fax messages.

$p$ be the percent of incoming call that involve fax messages.

Given that `$p=0.2$`

and `$n =20$`

. Thus `$X\sim B(20, 0.2)$`

.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{20}{x} (0.2)^x (1-0.2)^{20-x},\\ &\quad x=0,1,\cdots, 20 \end{aligned} $$ `

a. The probability that at most 8 of the calls involve a fax message is

` $$ \begin{aligned} P(X\leq 8) & =P(X\leq 8)\\ &= \sum_{x=0}^{8} P(x)\\ &= 0.99 \end{aligned} $$ `

b. The probability that exactly 8 of the calls involve a fax message is

` $$ \begin{aligned} P(X = 8) & =\binom{20}{8} (0.2)^{8} (1-0.2)^{20-8}\\\\ & = 0.022\\ \end{aligned} $$ `

c. The probability that at least 8 of the calls involve a fax message is

` $$ \begin{aligned} P(X\geq 8) & =1-P(X\leq 7)\\ &= 1-\sum_{x=0}^{7} P(x)\\ &= 1- 0.968\\ & = 0.032 \end{aligned} $$ `

d. The probability that more than 8 of the calls involve a fax message is

```
$$
\begin{aligned}
P(X >8) & =1-P(X\leq 8)\
&= 1-\sum_{x=0}^{8} P(x)\
&= 1- 0.99\
& = 0.01
\end{aligned}
$$</p>
<p>
```

#### Further Reading

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