Bayes' theorem

[edit] Bayesian Inference

Bayes' theorem is about conditional probabilities. Probability is about sets of outcomes. We start by assuming that these outcomes are equally likely. Suppose we have a bag full of balls, each ball is either red or blue. Each ball is also either Small or Big. Taking a ball from the bag is an outcome.

	Red	Blue	Total
Small	20	40	60
Big	10	30	40
Total	30	70	100

The conditional probability of a ball taken from the bag being Red if we already know it is Big is 10/40. This is written,

These are conditional probabilities. P(Red | Big) means,

 First I found that the ball was Big.  What then is the probability of it being red.

The probabilities for a a ball being red P(Red) is,

Note that P(Red | Big) has no meaning by itself. Instead probability has two sets,

• The set of events that register success.
• The domain from which those events are taken.

Note that,

P(Red) = P(Red | All).

The probabilities for a a ball being Big P(Big) is,

Now the probability of a ball being Red and Big is,

or,

so,

similarly,

so the result is,

This is Bayes' theorm usually written as,

it is also true that,

so,