Question

In a survey of 300 college graduates, 53% reported that they entered a profession closely related to their college major. If 9 of those survey subjects are randomly selected with replacement for a follow-up survey, what is the probability that 3 of them entered a profession closely related to their college major

Answer 1

0.1348 = 13.48% probability that 3 of them entered a profession closely related to their college major.

Explanation:

For each graduate, there are only two possible outcomes. Either they entered a profession closely related to their college major, or they did not. The probability of a graduate entering a profession closely related to their college major is independent of other graduates. This, coupled with the fact that they are chosen with replacement, means that 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.

53% reported that they entered a profession closely related to their college major.

This means that
`p = 0.53`

9 of those survey subjects are randomly selected

This means that
`n = 9`

What is the probability that 3 of them entered a profession closely related to their college major?

This is P(X = 3).

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

`P(X = 9) = C_(9,3).(0.53)^(3).(0.47)^(6) = 0.1348`

0.1348 = 13.48% probability that 3 of them entered a profession closely related to their college major.