Financial Risk Manager Part 1
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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.
Other
Community
NNikitesh
A
0.19
B
0.53
C
0.47
D
0.175
Explanation:
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.
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