# Bayes' Theorem

In probability, Bayes’ theorem describes the relationships that exist within an array of simple and conditional probabilities. The simple form of Bayes’ theorem is P(A|B) = P(B|A) x P(A)/P(B). P(A|B) presents the probabilities that A happens in condition B. By using Bayes’ theorem, people can find out how to change their mind after they get new evidence.

What does this theorem allow you to do that traditional statistics do not?
With Bayes’ theorem, people can analyze the situation with new evidence and response it.

What are some prerequisites for using Bayesian statistics?
Baye’s theorem is useful method when we want to find out the percentage of druggers in a group. Ever time the drugger being tested, the probability that a drugger in positive drug reaction (+) is 99% and the probability a non-dugger in negative drug reaction (-) is 99%. So drug test is not 100% precise.
For example, if we do the drug test in a group and we already know 0.5% people in this group are druggers. And we want to know the probability that who has positive drug reaction and is a drugger at the same time. We assume that “D” is a drugger, “N” is a non-drugger and “+” is someone who has positive drug reaction.
P(D) = 0.005
P(N) = 1 - 0.005 = 0.995
P(+|D) = a dugger has positive reaction = 0.99
P(+|N) = a non-drugger has positive reaction = 0.01
P(+) = P(+,D) + P(+,N)
= P(+|D)P(D) + P(+|N)P(N)
= (0.5% x 99%) + (99.5% x 1%)
= 0.495% + 0.995%
= 1.49%
= 0.0149
P(D|+) = P(+|D)*P(D) / P(+)
= P(+|D)*P(D) / [ P(+|D)P(D) + P(+|N)P(N) ]
= 0.99*0.005 / (0.99*0.005 + 0.01*0.995)
= 0.3322