# Solved (Free): A particular telephone number is used to receive both voice calls and fax messages

#### ByDr. Raju Chaudhari

Mar 13, 2021

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>