Can you trust your judgment with quick math calculations?


Suppose you undergo a medical test for a relatively rare cancer. The cancer has an incidence of 1 percent among the general population. Extensive trials have shown that the reliability of the test is 79 percent. More precisely, although the test does not fail to detect the cancer when it is present, it gives a positive result in 21 percent of the cases where no cancer is present - what is known as a false positive. When you are tested, the test produces a positive diagnosis. What is the probability that you have the cancer?
[The Unfinished Game by Keith Devlin page 140]

Spoiler:

If you are like most people, you will assume that if the test has a reliability rate of nearly 80 percent, and since you have tested positive, the likelihood that you do indeed have the cancier is about 80 percent (i.e., the probability is approximately 0.8). Are you right?


No. Given the scenario I just described, the likelihood that you have the cancer is just 4.6 percent (i.e., the probability is 0.046). There is a less-than-5-percent chance that you have the cancer.
...
Here is how you arrive at that figure:


P(H) = 0.01


P(E|H) = 1 (the test always shows positive if the cancer is present)


P(Hwrong) = 0.99 (99 percent of the population is cancer-free)


P(E|Hwrong) = 0.21 (the test gives a false positive in 21 percent of cases)


So by Bayes' formula
P(H|E) = 0.01*1 / ((0.01*1) + (0.99*0.21))
= 0.01 / 0.2179
= 0.0459
[The Unfinished Game by Keith Devlin pages 140 to 141]