Due to rising health insurance costs, 45 million people in the United States go without health...

Due to rising health insurance costs, 45 million people in the United States go without health insurance. Sample data representative of the national health insurance coverage for individuals 18 years of age and older are shown here.

Health Insurance

Age Yes No

18-34 .375 .085

35 and up .475 .065

a. If a randomly selected individual is 35 and older, what is the probability that she/he does not have health insurance coverage? Hint: Compute the marginal probabilities.

b. Are age and health insurance coverage (or lack of coverage) independent? Why? Use the proper formula to prove it.

c. If two unrelated individuals are randomly selected, what is the probability that "at least one" of them does not have health insurance?

Solutions

Expert Solution

The marginal probabilities are calculated as below.

Age	Health Insurance Yes	Health Insurance No	Total
18-34	0.375	0.085	0.46
35 and up	0.475	0.065	0.54
Total	0.85	0.15	1

Probability that she/he does not have health insurance coverage given randomly selected individual is 35 and older,

= Probability that she/he does not have health insurance coverage and individual is 35 and older / Probability that randomly selected individual is 35 and older

= 0.065 / 0.54

= 0.1203704

Probability that she/he does not have health insurance coverage and individual is 35 and older

= P(Health Insurance No AND Age 35 and up) = 0.065

Now,

P(Health Insurance No) = 0.15

P(Age 35 and up) = 0.54

P(Health Insurance No) * P(Age 35 and up) = 0.15 * 0.54 = 0.081

Since,

P(Health Insurance No AND Age 35 and up) P(Health Insurance No) * P(Age 35 and up)

age and health insurance coverage (or lack of coverage) are not independent.

Probability that "at least one" of them does not have health insurance = 1 - Probability that all of them have health insurance

= 1 - 0.85 * 0.85

= 0.2775

orchestra answered 1 year ago

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