Question

A health statistics agency in a certain country tracks the number of adults who have health insurance. Suppose according to the agency, the uninsured rates in a recent year are as follows: 5.3% of those under the age of 18, 12.6% of those ages 18–64, and 1.3% of those 65 and older do not have health insurance. Suppose 22.6% of people in the county are under age 18, and 62.1% are ages 18–64.
(a)
What is the probability that a randomly selected person in this country is 65 or older?
________
(b)
Given that a person in this country is uninsured, what is the probability that the person is 65 or older? (Round your answer to three decimal places.)
_______

Answer 1

Answer:

a) There is a 15.3% probability that a randomly selected person in this country is 65 or older.

b) Given that a person in this country is uninsured, there is a 2.2% probability that the person is 65 or older.

Step-by-step explanation:

We have these following percentages:

5.3% of those under the age of 18, 12.6% of those ages 18–64, and 1.3% of those 65 and older do not have health insurance.

22.6% of people in the county are under age 18, and 62.1% are ages 18–64.

(a) What is the probability that a randomly selected person in this country is 65 or older?

22.6% are under 18

62.10% are 18-64

The rest are above 65

So

100% - (22.6% + 62.10%) = 15.3%

There is a 15.3% probability that a randomly selected person in this country is 65 or older.

b) Given that a person in this country is uninsured, what is the probability that the person is 65 or older?

This can be formulated as the following problem:

What is the probability of B happening, knowing that A has happened.

It can be calculated by the following formula

$P = \frac{P(B).P(A/B)}{P(A)}$

Where P(B) is the probability of B happening, P(A/B) is the probability of A happening knowing that B happened and P(A) is the probability of A happening.

So, what is the probability that a person is 65 and older, given that the person is uninsured.

P(B) is the probability that a person is 65 and older. From a), we have that $P(B) = 0.153$

P(A/B) is the probability is uninsured, given that that person is 65 and older. So $P(A/B) = 0.013$

P(A) is the probability that a person is uninsured. That is the sum of 5.3% of 22.6%, 12.6% of 62.1% and 1.3% of 15.3%. So:

$P(A) = 0.053*(0.226) + 0.126*(0.621) + 0.013*(0.153) = 0.0922$

So

$P = \frac{P(B).P(A/B)}{P(A)} = \frac{0.153*0.013}{0.0922} = 0.022$

Given that a person in this country is uninsured, there is a 2.2% probability that the person is 65 or older.