Question

According to a survey, 78% of residents in Summerville own a home. If 30 residents

are randomly selected, find the probability exactly 24 residents own a home.

Answer 1

0.1732 = 17.32% probability exactly 24 residents own a home.

Explanation:

For each resident, there are only two possible outcomes. Either they own a home, or they do not. The probability of a resident owning a home is independent of any other resident. This means that the binomial probability distribution is used 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.

78% of residents in Summerville own a home.

This means that
`p = 0.78`

30 residents are randomly selected:

This means that
`n = 30`

Find the probability exactly 24 residents own a home.

This is P(X = 24).

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

`P(X = 24) = C_(30,24).(0.78)^(24).(0.22)^(6) = 0.1732`

0.1732 = 17.32% probability exactly 24 residents own a home.