Definition:Conditional Probability

Definition
Let $$\mathcal E$$ be an experiment with probability space $$\left({\Omega, \Sigma, \Pr}\right)$$.

Let $$A, B \in \Sigma$$ be events of $$\mathcal E$$.

We write the conditional probability of $$A$$ given $$B$$ as $$\Pr \left({A | B}\right)$$, and define it as:
 * the probability that $$A$$ has occurred, given that $$B$$ has occurred.

We have that $$\Pr \left({A | B}\right) = \frac {\Pr \left({A \cap B}\right)} {\Pr \left({B}\right)}$$.

This is derived as follows.

Suppose it is given that $$B$$ has occurred.

Then the probability of $$A$$ having occurred may not be $$\Pr \left({A}\right)$$ after all.

In fact, we can say that $$A$$ has occurred iff $$A \cap B$$ has occurred.

So, if we know that $$B$$ has occurred, the conditional probability of $$A$$ given $$B$$ is $$\Pr \left({A \cap B}\right)$$.

It follows then, that if we don't actually know whether $$B$$ has occurred or not, but we know its probability $$\Pr \left({B}\right)$$, we can say that:


 * The probability that $$A$$ and $$B$$ have both occurred is the conditional probability of $$A$$ given $$B$$ multiplied by the probability that $$B$$ has occurred.

Hence:
 * $$\Pr \left({A | B}\right) = \frac {\Pr \left({A \cap B}\right)} {\Pr \left({B}\right)}$$