Approximately 5% of workers in the US use public transportation to get to work. You randomly select 25 workers and ask if they use public transportation to get to work. Find the probability that exactly 2 workers say yes.

For each worker, there are only two possible outcomes. Either they say yes, or they say no. The probability of a worker saying yes is independent of any other worker, which 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.

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

And p is the probability of X happening.

5% of workers in the US use public transportation to get to work.

This means that [tex]p = 0.05[/tex]

You randomly select 25 workers

This means that [tex]n = 25[/tex]

Find the probability that exactly 2 workers say yes.

This is P(X = 2). So

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]P(X = 2) = C_{25,2}.(0.05)^{2}.(0.95)^{23} = 0.2305[/tex]

0.2305 = 23.05% probability that exactly 2 workers say yes.