Question

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

Answer 1

99.1% probability that at most 8 of the calls involve a fax message

Explanation:

For each call, there are only two possible outcomes. Either it is a fax message, or it is not. The probability of a call being a fax message is independent from other calls. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

Suppose that 20% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls

This means that
`p = 0.2, n = 20`

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

`P(X \leq 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)`

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(20,0).(0.2)^(0).(0.8)^(20) = 0.012`

`P(X = 1) = C_(20,1).(0.2)^(1).(0.8)^(19) = 0.058`

`P(X = 2) = C_(20,2).(0.2)^(2).(0.8)^(18) = 0.137`

`P(X = 3) = C_(20,3).(0.2)^(3).(0.8)^(17) = 0.205`

`P(X = 4) = C_(20,4).(0.2)^(4).(0.8)^(16) = 0.218`

`P(X = 5) = C_(20,0).(0.2)^(5).(0.8)^(15) = 0.175`

`P(X = 6) = C_(20,6).(0.2)^(6).(0.8)^(14) = 0.109`

`P(X = 7) = C_(20,7).(0.2)^(7).(0.8)^(13) = 0.055`

`P(X = 8) = C_(20,8).(0.2)^(8).(0.8)^(12) = 0.022`

`P(X \leq 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) = 0.012 + 0.058 + 0.137 + 0.205 + 0.218 + 0.175 + 0.109 + 0.055 + 0.022 = 0.991`

99.1% probability that at most 8 of the calls involve a fax message