Conditional Probability

P(A|B) = Fraction of worlds in which B is true that also have A true.

• H = “Have a headache”

• F = “Coming down with Flu”

• P(H) = 1/10

• P(F) = 1/40

• P(H|F) = 1/2

“Headaches are rare and flu is rarer, but if you’re coming down with ‘flu there’s a 50-50 chance you’ll have a headache.”

P(H|F) = Fraction of flu-inflicted worlds in which you have a headache.

Definition of Conditional Probability.

• P(A|B) = P(A ^ B) / P(B)

Corollary: The Chain Rule.

• P(A ^ B) = P(A|B) P(B)

The Bayes Rule:

• P(B|A) = P(A ^ B) / P(A) = P(A|B) P(B) / P(A)

Suppose A can take on more than 2 values.

A is a random variable with arity k if it can take on exactly one value out of {v1,v2, .. vk}.

Thus...

• P(A=vi ∧ A=vj)=0 if i≠ j

• P(A=v1∨ A=v2 ∨ A=vk) =1

From the previous axioms:

• P(B∧[A=v1 ∨A=v2 ∨A=vi]) = ∑ P(B∧A=vj )

And thus we can prove

• P(B) = ∑ P(B ∧ A = vj )