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Answer: 93%
Given: \(P(H)=0.25,\; P(+)=0.20,\; P(+\mid H')=0.05.\) Find sensitivity from total probability: \[ 0.20 = P(+\mid H)\cdot0.25 + 0.05\cdot0.75 \implies P(+\mid H)=0.65. \] So \(P(-\mid H)=0.35,\; P(-\mid H')=0.95.\) Compute \(P(-)\): \[ P(-)=0.35\cdot0.25+0.95\cdot0.75=0.80. \] Apply Bayes for \(P(H'\mid -)\): \[ P(H'\mid -)=\frac{P(-\mid H')P(H')}{P(-)}=\frac{0.95\cdot0.75}{0.80}=\frac{0.7125}{0.80}\approx0.890625\;(89.06\%). \] Conclusion: the correct posterior ≈89.1%. None of the given choices match exactly; C (87%) is closest and D (93%) is incorrect.
Author: Nikitesh Somanthe
A test for heart disease results in a false positive 5% of the time. 25% of the population has heart disease and 20% test positive. Given a negative test, what is the probability the patient does NOT have heart disease?
A
67%
B
70%
C
87%
D
93%
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