The Multiplication Rule

In probability theory, the Multiplication Rule states that the probability that $A$ and $B$ occur is equal to the probability that $A$ occurs times the conditional probability that $B$ occurs, given that we know $A$ has already occurred. This rule can be written:

$\displaystyle P(A \cap B) = P(B) \cdot P(A|B)$

Switching the role of $A$ and $B$, we can also write the rule as:

$\displaystyle P(A\cap B) = P(A) \cdot P(B|A)$

We obtain the general multiplication rule by multiplying both sides of the definition of conditional probability by the denominator. That is, in the equation $\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$, if we multiply both sides by $P(B)$, we obtain the Multiplication Rule. 

The rule is useful when we know both $P(B)$ and $P(A|B)$, or both $P(A)$ and $P(B|A).$

Example

Suppose that we draw two cards out of a deck of cards and let $A$ be the event the the first card is an ace, and $B$ be the event that the second card is an ace, then:

$\displaystyle P(A)=\frac { 4 }{ 52 }$

And:

$\displaystyle P\left( { B }|{ A } \right) =\frac { 3 }{ 51 }$

The denominator in the second equation is $51$ since we know a card has already been drawn. Therefore, there are $51$ left in total. We also know the first card was an ace, therefore:

$\displaystyle \begin{aligned} P(A \cap B) &= P(A) \cdot P(B|A)\\ &= \frac { 4 }{ 52 } \cdot \frac { 3 }{ 51 } \\ &=0.0045 \end{aligned}$

Independent Event

Note that when $A$ and $B$ are independent, we have that $P(B|A)= P(B)$, so the formula becomes $P(A \cap B)=P(A)P(B)$, which we encountered in a previous section. As an example, consider the experiment of rolling a die and flipping a coin. The probability that we get a $2$ on the die and a tails on the coin is $\frac{1}{6}\cdot \frac{1}{2} = \frac{1}{12}$, since the two events are independent.