Conditional Probability

probability-theory

Definition

Conditional Probability

Conditional probability is the probability of an event after restricting attention to the case where another event is known to occur.

In a probability space , let be events with . The conditional probability of given is

The event becomes the new effective sample space. Within that restricted world, happens exactly on the overlap .

Derivation

Conditioning on means replacing the original world by the smaller world . Any event outside is now irrelevant, because it contradicts the information that occurred.

For an event , the part of that remains possible after conditioning on is therefore

To turn this restricted probability into an ordinary probability again, the new world must have total probability . So we rescale all probabilities inside by the same normalising factor :

The normalisation condition is

Substituting gives

Since , this forces

Thus the probability of in the restricted world is

This restricted probability measure is written as , giving

Independence

If two events and are independent, i.e. , and , then:

Therefore: