Question

Dr Levin has a patient who is very sick. The only treatment available is a risky operation. The probability that the patient will survive the operation is 0.7.

Dr Levin finds out that there is a less risky procedure which provides information that predicts whether or not the patient will survive the operation.

The test has the following characteristics.

True-positive rate: The probability that the results of this test will be positive if the patient will survive the operation is 0.9

False-positive rate: The probability that the results of this test will be positive if the patient will not survive the operation is 0.2

True-negative rate: The probability that the results of this test will be negative if the patent will not survive the operation is 0.8

False-negative rate: The probability that the results of this test will be negative if the patient will survive the operation is 0.1

a)What is the patient’s probability of surviving the operation if the test is positive?

b)What is the patient’s probability of surviving the operation if the test is negative?

c)What is the patient’s probability of not surviving the operation if the test is positive?

d)What is the patient’s probability of not surviving the operation if the test is negative?

e)What is the unconditional probability that the test will be positive?

f)What is the unconditional probability that the test will be negative?

       Hint: Use Bayes’ Theorem

SHOW WORK

Answer 1

P(P | S) = 0.9
P(P | S') = 0.2
P(P' | S) = 0.8
P(P' | S') = 0.1

P(S) = 0.7 and P(S') = 0.3

a)
P(S | P) = P(P | S) * P(S) / (P(P | S) * P(S) + P(P | S') * P(S'))

= 0.9*0.7 / (0.9*0.7 + 0.2 * 0.3)
= 0.9130

b)
P(S | P') = P(P' | S)*P(S) / (P(P' | S)*P(S) + P(P' | S')*P(S'))
= 0.8*0.7/(0.8*0.7 + 0.1*0.3)
= 0.9492

c)
P(S' | P) = P(P|S')*P(S') / (P(P|S')*P(S') + P(P|S)*P(S))
= 0.2*0.3/(0.2*0.3 + 0.9*0.7)
= 0.0870

d)
P(S' | P') = P(P'|S')*P(S') / (P(P'|S')*P(S') + P(P'|S)*P(S))
= 0.1*0.3/(0.1*0.3 + 0.8*0.7)
= 0.0508

e)
P(P) = (0.9*0.7 + 0.2 * 0.3) = 0.6900

f)
P(P') = (0.8*0.7 + 0.1*0.3) = 0.59