## Use of Bayes’ Theorem in Epidemiology

Bayes’ theorem is arguably one of the most powerful and simple probabilistic equations that can be easily used in day to day life. Notably, it can overturn intuitive thoughts by revealing the actual probabilities behind seemingly easy problems.

For example: Say a patient is being tested by a doctor. The doctor has a test for a disease that has a 99% accuracy rate. 99% of patients who are sick test positive, and 99% of patients who are healthy test negative. There is also data that 1% of people in the country, which has a population of 10,000, who have the disease. At first glance, it would appear that this is an extremely reliable indicator of patient status, and diagnosis should be almost entirely trusted using this test. However, the reality betrayed by Bayes’ theorem tells a drastically different story.

Let’s plug the numbers in. Bayes’ theorem is: P(A|B) = P(A)P(B|A)/P(B).

If we set P(A|B) to mean the chance of having the disease given you test positive, then P(A) is the chance of having the disease, .01. The accuracy of the test is 99%, so the chance of testing positive if you have the disease, i.e. P(B|A) = .99. Because we know there is a 1% error rate in a country of 10,000, there will be 198 people who test positive, even though only 100 have the disease. Therefore P(B) = 198/10,000, or 0.0198.

P(A|B) = (.99 * .01)/0.0198 = .5 or 50%. Therefore, the chance of having the disease if you test positive on a 99% accurate test is the flip of a coin.

In real life, tests are much less accurate than the near-ideal of 99%. As we can see from the low chances of this test, a holistic round of tests is required in order to create a certainty regarding a patient’s status. This example shows the use of Baye’s theorem illuminates the probabilistic reality lurking behind what appears to be an easy and intuitive answer.

Example sourced from http://www.scientificamerican.com/article/what-is-bayess-theorem-an/ and http://sphweb.bumc.bu.edu/otlt/MPH-Modules/BS/BS704_Probability/BS704_Probability6.html