Question

Which of the following relations hold true for two independent events A and B?

• A. $P(A\cup B)=P\left(A\right)+P\left(B\right)$
• B. $P(A\cap B)=0$
• C. $P(A\cup B)=P\left(A\right).P\left(B\right)$
• D. $P(A\cap B)=P\left(A\right).P\left(B\right)$

Answer 1

The correct option is D $P(A\cap B)=P\left(A\right).P\left(B\right)$

We saw that for two independent events A and B, $P(A\cap B)=P\left(A\right)P\left(B\right)$ . But how do we get this relation?
For that, let us consider the definition of independent events. Two events are said to be independent if the occurrence of one does not change the probability of the occurrence or nonoccurrence of other. So, the probability of A given B ,$P\left(\frac{A}{B}\right)=P\left(A\right)$. (If you are not familiar with this, you can look at conditional probability)
But we have $P\left(\frac{A}{B}\right)=\frac{P(A\cap B)}{P\left(B\right)}$
For two independent events A and B, $P\left(A\right)=\frac{P(A\cap B)}{P\left(B\right)}$
$\Rightarrow P(A\cap B)=P\left(A\right)P\left(B\right)$