Question

It is known that 50% of adult workers have a high school diploma. If a random sample of 8 adult workers is selected, what is the probability that less than 6 of them have a high school diploma

Answer 1

85.56% probability that less than 6 of them have a high school diploma

Explanation:

For each adult, there are only two possible outcomes. Either they have a high school diploma, or they do not. The probability of an adult having a high school diploma is independent of other adults. 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.

50% of adult workers have a high school diploma.

This means that
`p = 0.5`

If a random sample of 8 adult workers is selected, what is the probability that less than 6 of them have a high school diploma

This is P(X < 6) when n = 8.

`P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)`

In which

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

`P(X = 0) = C_(8,0).(0.5)^(0).(0.5)^(8) = 0.0039`

`P(X = 1) = C_(8,1).(0.5)^(1).(0.5)^(7) = 0.0313`

`P(X = 2) = C_(8,2).(0.5)^(2).(0.5)^(6) = 0.1094`

`P(X = 3) = C_(8,3).(0.5)^(3).(0.5)^(5) = 0.2188`

`P(X = 4) = C_(8,4).(0.5)^(4).(0.5)^(4) = 0.2734`

`P(X = 5) = C_(8,5).(0.5)^(5).(0.5)^(3) = 0.2188`

`P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) = 0.0039 + 0.0313 + 0.1094 + 0.2188 + 0.2734 + 0.2188 = 0.8556`

85.56% probability that less than 6 of them have a high school diploma