In a particular region, 4% of the population is thought to have a certain diseas

Question

In a particular region, 4% of the population is thought to have a certain disease. A standard diagnostic test has been found to correctly identify 95% of the people who have the disease. However, the test also incorrectly diagnoses 11% of those who do not have the disease as having the disease (in other words, the person does not have the disease but the test tells them that they do).

1. What is the probability the test comes back positive?

2. What is the probability the test comes back positive and the person actually has the disease?

3. If the test comes back positive, what then is the conditional probability that the person actually does have the disease?

Explanation / Answer

Let

D = has disease
P = tests positive

1.

Hence, by Bayes' Rule,

P(P) = P(D) P(P|D) + P(D') P(P|D') = 0.04*0.95 + (1-0.04)*0.11 = 0.1436 [ANSWER]

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2.

Note that

P(P n D) = P(D) P(P|D) = 0.04*0.95 = 0.038 [ANSWER]

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3.

Thus,

P(D|P) = P(P n D)/P(P) = 0.038/0.1436 = 0.264623955 [ANSWER]

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