A screening test for a certain disease is prone to giving false positives or fal

Question

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is 0.13. If the patient does not have the disease, the probability that the test indicates a (false) positive is 0.10. Assume that 3% of the patients being tested actually have the disease. Suppose at one patient is chosen at random and tes (a) this patient tests positive given that he/she has the disease (b) this patient does not have disease and test positive. (c) this patient has the disease and the test positive. (d) this patient tests positive. (e) this patient has the disease given that he/she tests positive.

Explanation / Answer

The probability that the patient has the disease is 0.03.
The probability that the patient does not have the disease is 0.97.

#1: The probability that the patient has the disease and tests positive is 0.03 * 0.87, or 0.0261.
#2: The probability that the patient has the disease but tests negative is 0.03 * 0.13, or 0.0039.
#3: The probability that the patient does not have the disease but tests positive is 0.97 * 0.10, or 0.097.
#4: The probability that the patient does not have the disease and tests negative is 0.97 * 0.90, or 0.873.

So the answers are:
a) #1, which is 0.0261.
b) #3, which is 0.097.
c) #1 + #3, which is 0.0261 + 0.097, or 0.1231.
d) #1 / (#1 + #3) = 0.0261 / 0.1231 = 0.2120.

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