Multiplication Rule

Definition:

The definition of conditional probability in equation can be rewritten to provide a general expression for the probability of the intersection of two events. This formula is referred to as a multiplication rule for probabilities.

\[P\left( A\cap B \right)\text{ }=\text{ }P\left( \frac{B}{A} \right)P\left( A \right)\text{ }=\text{ }P\left( \frac{A}{B} \right)P\left( B \right)\]

The last expression in above Equation is obtained by interchanging $A$ and $B$.

Example:

The probability that an automobile battery subject to high engine compartment temperature suffers low charging current is $0.7$. The probability that a battery is subject to high engine compartment temperature is $0.05$.

Let $C$ denote the event that a battery suffers low charging current, and let $T$ denote the event that a battery is subject to high engine compartment temperature. The probability that a battery is subject to low charging current and high engine compartment temperature is

\[P\left( C\cap T \right)\text{ }=\text{ }P\left( \frac{C}{T} \right)P\left( T \right)\text{ }=\text{ }0.7\text{ }\times \text{ }0.05\text{ }=\text{ }0.035\]