A life insurance company plans to assess its future liabilities by sampling the

Question

A life insurance company plans to assess its future liabilities by sampling the records of its policyholders. A pilot study has turned up three clients---one living in Alaska, one in Missouri, and one in Vermont---whose estimated chances of surviving to the year 2025 are 0.7, 0.9 and 0.3, respectively. It is reasonable to assume that life expectancies of the three clients are mutually independent. What is the probability that by the end of 2024 the company will have to pay death benefits to exactly one of the three? at least one of the three? The following is known about a colormetric method used to test lake water for nitrates. If water specimens contain nitrates, a solution dropped into the water will cause the specimen to turn red 95% of the time. When used on water specimens without nitrates, the solution causes the water to turn red 10% of the time (because chemicals other than nitrates are sometimes present and they also react to the agent). Past experience in a lab indicates that nitrates are contained in 30% of the water specimens that are sent to the lab for testing. If a water specimen is randomly selected from among those sent to the lab, what is the probability that it will turn red when tested? If a water specimen is randomly selected and turns red when tested, what is the probability that it actually contains nitrates?

Explanation / Answer

Let
A denote that the water sample contains nitrates ,
and B denote the sample turns red

P[A & B] = 30% * 95% = .285 = 28.5%
P[A' & B] = 70% * 10% = .07 = 7%

(a) P[B] = P[A & B] + P[A' & B] = 28.5% + 7% = 35.5%

(b) P[A | B] = 28.5 / 35.5 = 0.8028 or 80.28%

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