The law of multiplication in probability states that the probability of two independent events occurring simultaneously is equal to the product of their individual probabilities.

When we have two independent events, denoted as $A$ and $B$, the probability of both events occurring together can be mathematically expressed as:

To illustrate this concept, let’s consider an example involving the probability of rolling a $4$ on a fair six-sided die and flipping a head on a fair coin. The probability of rolling a $4$ is $\frac{1}{6}$, while the probability of flipping a head is $\frac{1}{2}$. Since these two events are independent, the probability of both rolling a $4$ and flipping a head is calculated as follows:

The law of multiplication can also be extended to encompass more than two independent events. For instance, if we have three independent events, $A$, $B$, and $C$, the probability of all three events occurring together is given by:

It is crucial to understand that the law of multiplication applies only to independent events. In cases where events are dependent, calculating the probability of both events occurring together becomes more complex and typically involves the use of conditional probabilities.