Question

In a recent survey, 66% of the community favored building a health center in their neighborhood. Suppose 12 citizens are randomly chosen and asked if they favor building the health center. Assume all requirements for computing a binomial probability are met. What is the probability that exactly 10 of the 12 individuals favor building the health center? (Round your answer to 4 decimal places)

Answer 1

0.1197 = 11.97% probability that exactly 10 of the 12 individuals favor building the health center

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.

66% of the community favored building a health center in their neighborhood.

This means that
`p = 0.66`

12 citizens

This means that
`n = 12`

What is the probability that exactly 10 of the 12 individuals favor building the health center?

This is P(X = 10).

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

`P(X = 10) = C_(12,10).(0.66)^(10).(0.34)^(2) = 0.1197`

0.1197 = 11.97% probability that exactly 10 of the 12 individuals favor building the health center