Chain Rule for Probability

Theorem

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$ such that $\map \Pr B > 0$.

$\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$

Suppose it is given that $B$ has occurred.

Then the probability of $A$ having occurred may not be $\map \Pr A$ after all.

In fact, we can say that $A$ has occurred if and only if $A \cap B$ has occurred.

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

It follows then, that:

if we don't actually know whether $B$ has occurred or not

but we know its probability $\map \Pr B$

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:

$\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$

$\blacksquare$

The Chain Rule for Probability can also be presented in the form:

$\condprob A B \cdot \map \Pr B = \map \Pr {A \cap B}$