Bayes' Theorem

Demonstration of Bayes' Law
You can change the proportions and see the results (hit return or go to another field after changing the value).

The base rate, the proportion of people have Disease X is:
The true positive rate is:
The false positive rate is:

Question: If a person is tests positive, what is the probability the person has the disease?

Expected frequencies for 10,000 people.

198 + 882 = 1080 test positive.
198/1080 = 0.1833 of those testing positive have the disease.

Therefore, the probability that one has the disease given that they test positive P(D|T) is:

0.1833.

The same answer can be computed with Bayes's Theorem:

P(D|T) = Probability of having the disease if you tested positive.
P(T|D) = Probability of testing positive if you have the disease.
P(T|D') = Probability of testing postive if you do not have the disease.
P(D) = Probability of the disease.
P(D') = Probability of not having the disease.

For this example:

P(T|D) = 0.99
P(T|D') = 0.09
P(D) = 0.02
P(D') = 0.98