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 $\Pr \left({A \mid B}\right)$, and define it as:

the probability that $A$ has occurred, given that $B$ has occurred.

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

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:

$\Pr \left({A \mid B}\right) = \dfrac {\Pr \left({A \cap B}\right)} {\Pr \left({B}\right)}$

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.6 \ (19)$