Financial Risk Manager Part 1
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A test for heart disease results in a correct positive diagnosis 95% of the time and a correct negative diagnosis 99% of the time. 25% of the population has heart disease. What is the probability of a positive test?
Other
Community
NNikitesh
A
5%
B
15%
C
25%
D
95%
Explanation:
Let's define the following events:
- H = heart disease
- H' = no heart disease
-
- = positive test
-
- = negative test
Given:
- P(+|H) = 0.95 (correct positive diagnosis)
- P(-|H') = 0.99 (correct negative diagnosis)
- P(H) = 0.25 (25% of population has heart disease)
- P(H') = 0.75 (75% of population has no heart disease)
We can also derive:
- P(-|H) = 1 - P(+|H) = 1 - 0.95 = 0.05 (false negative)
- P(+|H') = 1 - P(-|H') = 1 - 0.99 = 0.01 (false positive)
Using the law of total probability: P(+) = P(+|H) × P(H) + P(+|H') × P(H') P(+) = 0.95 × 0.25 + 0.01 × 0.75 P(+) = 0.2375 + 0.0075 = 0.245 or 24.5%
Rounding to the nearest percentage gives 25%, which matches option C.
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