Bayes Theorem: Conditional Probabilities---3) Bayes Law

• Now we define conditional probability (read as probability of S given A has occurred) as .

• Note this is a probability according to axioms, if we restrict ``bounding space'' Y to be A.

• Then, in terms of frequencies,

  

  whence

  

  which is Bayes Law.

• Of course Equation (3) is obvious from definition: we didn't need frequencies to prove it.