Part One: Probability

Conditional Probability

The Conditional Probability Triplet

A conditional probability is a modified probability - it incorporates the relationship between a pair of events. We define the quantity Pr{ A | B } as the probability that event A is observed in a single trial of our experiment, given that event B occurs on that same trial.

The formula is simple:

Pr{ A | B }=Pr{ A and B } / Pr{ B }.

Consider some algebra:

This is a general form of a product rule for pairs of events:

Pr{ A and B } = Pr{ A | B }*Pr{ B }.

In the special case of independent events, we have that:

Pr{ A | B } = Pr{ A };

Pr{ B | A } = Pr{ B };

Pr{ A and B } = Pr{ A }*Pr{ B }.