Financial Risk Manager Part 1
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A human health organization tracked a group of individuals for five years. At the commencement of the study, 25% were categorized as heavy smokers, 40% as light smokers, and the remaining as nonsmokers. Results revealed that light smokers were twice as likely as nonsmokers to die during the half-decade study, but only half as likely as heavy smokers. During the period, a randomly selected group member passed on. Compute the probability that the individual who died was a heavy smoker.
Explanation:
Explanation
This is a classic Bayes' Theorem problem. Let's define the events:
- H: Heavy smoker
- L: Light smoker
- N: Non-smoker
- D: Died during the study
Given probabilities:
- P(H) = 25% = 0.25
- P(L) = 40% = 0.40
- P(N) = 1 - 0.25 - 0.40 = 0.35
Conditional death probabilities:
Let P(D|N) = x
- P(D|L) = 2x (twice as likely as nonsmokers)
- P(D|H) = 4x (since light smokers are half as likely as heavy smokers, heavy smokers are twice as likely as light smokers)
Total probability of death:
P(D) = P(D|H)P(H) + P(D|L)P(L) + P(D|N)P(N) = (4x)(0.25) + (2x)(0.40) + (x)(0.35) = x(1.0 + 0.8 + 0.35) = 2.15x
Applying Bayes' Theorem:
P(H|D) = [P(D|H)P(H)] / P(D) = (4x × 0.25) / (2.15x) = (1.0x) / (2.15x) = 1.0 / 2.15 ≈ 0.4651 ≈ 47%
Therefore, the probability that a randomly selected individual who died was a heavy smoker is approximately 47%.