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Answer: 0.47
Let: D = Event of death L = Event of light smoker H = Event of heavy smoker N = Event of nonsmoker We need to calculate P(H | D) Given probabilities: P(H) = 0.25 (25% heavy smokers) P(L) = 0.40 (40% light smokers) P(N) = 0.35 (remaining 35% nonsmokers) Given relationships: P(D | L) = 2P(D | N) (light smokers twice as likely to die as nonsmokers) P(D | L) = 1/2P(D | H) (light smokers half as likely to die as heavy smokers) From the second relationship: P(D | H) = 2P(D | L) Applying Bayes' theorem: P(H | D) = [P(H) × P(D | H)] / [P(H) × P(D | H) + P(L) × P(D | L) + P(N) × P(D | N)] Substitute: = [0.25 × 2P(D | L)] / [0.25 × 2P(D | L) + 0.40 × P(D | L) + 0.35 × (1/2)P(D | L)] = [0.5P(D | L)] / [0.5P(D | L) + 0.4P(D | L) + 0.175P(D | L)] = 0.5P(D | L) / [1.075P(D | L)] = 0.5 / 1.075 ≈ 0.4651 ≈ 0.47 Therefore, the probability that the individual who died was a heavy smoker is approximately 0.47.
Author: Nikitesh Somanthe
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A human health organization tracked a group of individuals for 5 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.
A
0.19
B
0.53
C
0.47
D
0.175