Question

A sociologist is analyzing Social Security data. The number of individuals in the sample is 5. The probability that an individual receives Social Security is 20%. What is the probability that exactly 3 of the 5 individuals receive Social Security? 0.0512 0.0305 0.6125 0.1024

Answer 1

0.0512

Explanation:

For each individual, there are only two possible outomes. Either he receives Social Security, or he does not. The probability of an individual receiving Social Security is independent of other individuals. 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.

The number of individuals in the sample is 5.

This means that
`n = 5`

The probability that an individual receives Social Security is 20%.

This means that
`p = 0.2`

What is the probability that exactly 3 of the 5 individuals receive Social Security?

This is P(X = 3).

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

`P(X = 3) = C_(5,3).(0.2)^(3).(0.8)^(2) = 0.0512`