Question

In a study by Peter D. Hart Research Associates for the Nasdaq Stock Market, it was determined that 20% of all stock investors are retired people. In addition, 40% of all U.S. adults invest in mutual funds. Suppose a random sample of 25 stock investors is taken. a. What is the probability that exactly seven are retired people

Answer 1

0.0545 = 5.45% probability that exactly seven are retired people.

Explanation:

For each stock investor, there are only two possible outcomes. Either they are retired people, or they are not. Stock investors are independent. This means that 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.

20% of all stock investors are retired people.

This means that
`p = 0.2`

a. What is the probability that exactly seven are retired people?

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

`P(X = 7) = C_(20,7).(0.2)^(7).(0.8)^(13) = 0.0545`

0.0545 = 5.45% probability that exactly seven are retired people.