Question

Forty percent of all high school graduate work during the summer to earn money for college tuition for the upcoming fall term. Assuming a binomial distribution, if 6 graduates are selected at random, what is the probability that at least 2 graduates have a summer job

Answer 1

0.7667 = 76.67% probability that at least 2 graduates have a summer job

Explanation:

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.

Forty percent of all high school graduate work during the summer to earn money for college tuition for the upcoming fall term.

This means that
`p = 0.4`

6 graduates.

This means that
`n = 6`

What is the probability that at least 2 graduates have a summer job

This is:

`P(X \geq 2) = 1 - P(X < 2)`

In which

`P(X < 2) = P(X = 0) + P(X = 1)`. So

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

`P(X = 0) = C_(6,0).(0.4)^(0).(0.6)^(6) = 0.0467`

`P(X = 1) = C_(6,1).(0.4)^(1).(0.6)^(5) = 0.1866`

`P(X < 2) = P(X = 0) + P(X = 1) = 0.0467 + 0.1866 = 0.2333`

`P(X \geq 2) = 1 - P(X < 2) = 1 - 0.2333 = 0.7667`

0.7667 = 76.67% probability that at least 2 graduates have a summer job