Question

A diagnostic test for a certain disease is said to be 90% accurate in that, if a person has the disease, the test will detect it with a probability of 0.9. Also, if a person does not have the disease, the test will report that he or she does not have it with a probability of 0.9. Only 1% of the population has the disease in question. If a person is chosen at random from the population and the diagnostic test indicates that she has the disease, what is the conditional probability that she does in fact have the disease?

Answer 1

Answer:

Given,

To determine the conditional probability

consider,

P(Disease) = 0.01

P(No disease) = 1 - 0.01 = 0.99

P(Positive | Disease) = 0.9

P(Negative | No Disease) = 0.9

P(Positive| No Disease) = 1 - 0.9 = 0.1

now by using the Baye's Formula:

P(Disease | Positive) = (0.01*0.9) /(0.01 + 0.9 + 0.99 + 0.1)

= 1 / 12

So P(Disease | Positive) = 0.0833

Yes, we can say that the answer is surprising since even if the test comes positive(+) here there are only 8.33% of chance that she will actually has the disease.