Question

Answer the following questions and round your answers to 2 decimal places. 84% of all Americans live in cities with population greater than 50,000 people. If 50 Americans are selected at random, Find the probability that Exactly 42 of them live in cities with population greater than 50,000 people.

Answer 1

15.23% probability that exactly 42 of them live in cities with population greater than 50,000 people.

Explanation:

For each American, there are only two possible outcomes. Either they live in cities with population greater than 50,000 people. Or they do not. The probability of a person living in a city with population greater than 50,000 people is independent of any other person. 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.

84% of all Americans live in cities with population greater than 50,000 people.

This means that
`p = 0.84`

If 50 Americans are selected at random, Find the probability that Exactly 42 of them live in cities with population greater than 50,000 people.

This is P(X = 42) when n = 50. So

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

`P(X = 42) = C_(50,42).(0.84)^(42).(0.16)^(8) = 0.1523`

15.23% probability that exactly 42 of them live in cities with population greater than 50,000 people.