Question

A study indicates that 62% of students have have a laptop. You randomly sample 8 students. Find the probability that between 4 and 6 (including endpoints) have a laptop.

Answer 1

72.69% probability that between 4 and 6 (including endpoints) have a laptop.

Explanation:

For each student, there are only two possible outcomes. Either they have a laptop, or they do not. The probability of a student having a laptop is independent from 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.

A study indicates that 62% of students have have a laptop.

This means that
`n = 0.62`

You randomly sample 8 students.

This means that
`n = 8`

Find the probability that between 4 and 6 (including endpoints) have a laptop.

`P(4 \leq X \leq 6) = P(X = 4) + P(X = 5) + P(X = 6)`

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

`P(X = 4) = C_(8,4).(0.62)^(4).(0.38)^(4) = 0.2157`

`P(X = 5) = C_(8,5).(0.62)^(5).(0.38)^(3) = 0.2815`

`P(X = 6) = C_(8,6).(0.62)^(6).(0.38)^(2) = 0.2297`

`P(4 \leq X \leq 6) = P(X = 4) + P(X = 5) + P(X = 6) = 0.2157 + 0.2815 + 0.2297 = 0.7269`

72.69% probability that between 4 and 6 (including endpoints) have a laptop.