Question

If $A$ and $B$ are two events such that $P(A\cup B)=P(A\cap B),$ then which of the following statements(s) is/are always correct ?

• A. $A$ and $B$ are equally likely.
• B. $P(A\cap \overline{B})=0$
• C. $P(\overline{A}\cap B)=0$
• D. $P\left(A\right)+P\left(B\right)=1$

Answer 1

Answer: (A), (B), (C)

$P(\overline{A}\cap B)=0$

Given :
$P(A\cup B)=P(A\cap B)\Rightarrow P\left(A\right)+P\left(B\right)-P(A\cap B)=P(A\cap B)\Rightarrow 2P(A\cap B)=P\left(A\right)+P\left(B\right)$
This is only possible if , $P\left(A\right)=P\left(B\right)$
$\Rightarrow P(A\cap B)=P\left(A\right)=P\left(B\right)\cdots \left(i\right)$
So, $A,B$ are equally likely.
Now,
$P(A\cap \overline{B})=P\left(A\right)-P(A\cap B)=P\left(A\right)-P\left(A\right)=0$
[using $\left(i\right)$]
Similarly, $P(\overline{A}\cap B)=P\left(B\right)-P(A\cap B)=0$
and $P\left(A\right)+P\left(B\right)=2P\left(A\right)$ can be zero.
So, $P\left(A\right)+P\left(B\right)$ is not always equal to $1.$