Bayes' theorem in 1 minute

Where Tistou proves Bayes' rule, introduces prior and posterior probabilities, and finally goes to the beach.

What is the probability of events A and B to occur together? We disregard the underlying timing or causation link and don't give to A and B any specific role. The answer is both:

• The probability of A to happen times the probability of B knowing that A has happened.
• The probability of B to happen times the probability of A knowing that B has happened.

I can give some examples over a green juice.

This gives this beautiful formula:

P(A | B) × P(B) = P(A ∩ B) = P(B | A) × P(A)

Now if P(B) is not zero, we can divide both left and right members by this term to obtain:

P(A | B) = P(B | A) × P(A) / P(B)     Bayes' law.

In this setting, we know B occurred and get a refined probability that A will occur. We call:

• P(A | B): posterior probability
• P(B | A): likelihood (of event B occurring given that A is true).
• P(A): prior probability
• P(B): evidence

See, we knew P(A) (prior), and with the information that B occurred (evidence) we can have a more accurate probability than A will occur (posterior).

Cheers!

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