Question

According to a 2014 Gallup poll, 56% of uninsuredAmericans who plan to get health insurance say they will do sothrough a government health insurance exchange.1) What is the probability that in a random sample of 10people exactly 6 plan to get health insurance through agovernment health insurance exchange

Answer 1

24.27% probability that in a random sample of 10 people exactly 6 plan to get health insurance through a government health insurance exchange

Explanation:

For each uninsured American, there are only two possible outcomes. Either they plan to get insurance through a government health insurance exchange, or they do not. The probability of an adult planning to get insurance through a government health insurance exchange is independent from other adults. 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.

56% of uninsuredAmericans who plan to get health insurance say they will do sothrough a government health insurance exchange.

This means that
`p = 0.56`

What is the probability that in a random sample of 10people exactly 6 plan to get health insurance through agovernment health insurance exchange

This is P(X = 6) when n = 10.

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

`P(X = 10) = C_(10,6).(0.56)^(6).(0.44)^(4) = 0.2427`

24.27% probability that in a random sample of 10 people exactly 6 plan to get health insurance through a government health insurance exchange