Question

It a recent study of college students indicated that 30% of all college students had at least one tattoo.A small private college decided to randomly and independently sample 15 of their students andask if they have a tattoo. Find the probability that exactly 5 of the students reported that they didhave at least one tattoo.

Answer 1

20.61% probability that exactly 5 of the students reported that they did have at least one tattoo.

Explanation:

For each student, there are only two possible outcomes. Either they have at least one tattoo, or they do not. The probability of a student having at least one tattoo is independent of other students. So we use the binomial probability distibution 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.

30% of all college students had at least one tattoo

This means that
`p = 0.3`

Sample of 15 students

This means that
`n = 15`

Find the probability that exactly 5 of the students reported that they didhave at least one tattoo.

This is P(X = 5).

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

`P(X = 5) = C_(15,5).(0.3)^(5).(0.7)^(10) = 0.2061`

20.61% probability that exactly 5 of the students reported that they did have at least one tattoo.