Multiplication Theorem on Probability - Practice Questions & MCQ

Let $A$ and $B$ are two events associated with a sample space $S$. The set $A \cap B$ denotes the event that both $A$ and $B$ have occurred. In other words, $A \cap B$ denotes the simultaneous occurrence of the events $A$ and $B$. The event $A \cap B$ is also written as $A B$.

The probability of event $A B$ or $A \cap B$ can be obtained by using the conditional probability.

From the conditional probability of event A given that B has occurred is denoted by $\mathrm{P}(\mathrm{A} \mid \mathrm{B})$ and is given by

$
\mathrm{P}(\mathrm{~A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{~A} \cap \mathrm{~B})}{\mathrm{P}(\mathrm{~B})}, \mathrm{P}(\mathrm{~B}) \neq 0
$
Using this result, we can write

$
\mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=\mathrm{P}(\mathrm{~B}) \cdot \mathrm{P}(\mathrm{~A} \mid \mathrm{B})
$
Also, we know that

$
\begin{aligned}
\mathrm{P}(\mathrm{~B} \mid \mathrm{A}) & =\frac{\mathrm{P}(\mathrm{~B} \cap \mathrm{~A})}{\mathrm{P}(\mathrm{~A})}, \mathrm{P}(\mathrm{~A}) \neq 0 \\
\text { or } \quad \mathrm{P}(\mathrm{~B} \mid \mathrm{A}) & =\frac{\mathrm{P}(\mathrm{~A} \cap \mathrm{~B})}{\mathrm{P}(\mathrm{~A})}, \mathrm{P}(\mathrm{~A}) \neq 0 \quad(\because \mathrm{~A} \cap \mathrm{~B}=\mathrm{B} \cap \mathrm{~A})
\end{aligned}
$
Thus, $\quad \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B} \mid \mathrm{A})$
Combining (1) and (2), we get

$
\begin{aligned}
\mathrm{P}(\mathrm{~A} \cap \mathrm{~B}) & =\mathrm{P}(\mathrm{~A}) \cdot \mathrm{P}(\mathrm{~B} \mid \mathrm{A}) \\
& =\mathrm{P}(\mathrm{~B}) \cdot \mathrm{P}(\mathrm{~A} \mid \mathrm{B}) \quad(\text { provided } \mathrm{P}(\mathrm{~A}) \neq 0 \text { and } \mathrm{P}(\mathrm{~B}) \neq 0)
\end{aligned}
$

The above result is known as the multiplication rule of probability.

Multiplication rule of probability for more than two events
If $A, B$ and $C$ are three events associated with sample space, then we have

$
\begin{aligned}
P(A \cap B \cap C) & =P(A) P(B \mid A) P(C \mid A \cap B) \\
& =P(A) P(B \mid A) P(C \mid A B)
\end{aligned}
$

Similarly, the multiplication rule of probability can be extended for four or more events.