Bayes Theorem

statistics&probability

Bayes Theorem

Basic Theory

Bayes Theorem tells what is the probability of A given B:

bayes

Here is the simple proof of Bayes Theorem. We know that the probability of A given B, or the probability of B given A is like this:

bayes_1

Therefore we know that:

bayes_2

Simple Example

A typical problem solved by Bayes Theorem is drug testing problem:

1
Suppose that a drug test will produce 99% true positive results for drug users and 99% true negative results for non‐drug users. Suppose that 0.5% of people are users of the drug. What is the probability that a randomly selected individual with a positive test is a drug user?

For this problem, we know that:

bayes_3

Then we can calculate the result like this:

bayes_4