Question

According to a study, 33% of college students call their families each week. A social scientist, who studies communication, would like to study communication between college students and their family members. If the social scientist selects 60 college students, use a calculator to find the probability that of those 60 college students, between 18 and 20 of them call their families weekly. Round your answers to two decimal places.

Answer 1

31.49% probability that of those 60 college students, between 18 and 20 of them call their families weekly.

Explanation:

For each student, there are only two possible outcomes. Either they call their families each week, or they do not. The probability of a student calling their family each week is independent of other students. 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.

33% of college students call their families each week.

This means that
`p = 0.33`

60 college students

This means that
`n = 60`

Probability that of those 60 college students, between 18 and 20 of them call their families weekly.

`P(18 \leq X \leq 20) = P(X = 18) + P(X = 19) + P(X = 20)`

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

`P(X = 18) = C_(60,18).(0.33)^(18).(0.67)^(42) = 0.0988`

`P(X = 19) = C_(60,19).(0.33)^(19).(0.67)^(41) = 0.1075`

`P(X = 20) = C_(60,20).(0.33)^(20).(0.67)^(40) = 0.1086`

`P(18 \leq X \leq 20) = P(X = 18) + P(X = 19) + P(X = 20) = 0.0988 + 0.1075 + 0.1086 = 0.3149`

31.49% probability that of those 60 college students, between 18 and 20 of them call their families weekly.