Question 10.2: A medical lab is 99% effective in detecting a certain diseas......
A medical lab is 99% effective in detecting a certain disease when it is actually present, and it yields a false positive for 1% of healthy people tested. If 0.25% of the population actually have the disease, what is the probability that a person has the disease if the patient’s blood test is positive?
Let T be the event that the patient’s test result is positive, and D the event that the tested person has the disease. Then the desired probability is P(D|T) and is given by
P(D\mid T)=\frac{P(DT)}{P(T)}=\frac{{P}({T}\mid{D}){P}({D})}{{P}({T}\mid{D}){P}({D})+{P}({T}\mid{D}^{c}){P}({D}^{c})}= \frac{0.99*0.0025}{0.99*0.0025+0.01\ *0.9975}= 0.1988
The reason that only approximately 20% of the population whose test results are positive actually have the disease may seem surprising, but it is explained by the low incidence of the disease (just one person out of every 400 tested will have the disease and the test will correctly confirm that 0.99 have the disease, but the test will also state that 399 ∗ 0.01 = 3.99 have the disease and so the proportion of time that the test is correct is {}^{0.99} /_{0.99+3.99} = 0.1988.