Question

If $P(A\cup B)=P(A\cap B)$ for any two events A and B. then

• A. $P\left(A\right)=P\left(B\right)$
• B. $P\left(A\right)>P\left(B\right)$
• C. $P\left(A\right)<P\left(B\right)$
• D. None of these

Answer 1

The correct option is A

$P\left(A\right)=P\left(B\right)$

We know that,

$P(A\cup B)+P\left(B\right)-P(A\cap B)$

$\Rightarrow P(A\cap B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\left[P\right(A\cup B)=P(AcapB\left)\right]$

$\Rightarrow P\left(A\right)-P(a\cap B)+P\left(B\right)-P(A\cap B)=0$ ....... (1)

But,

$P\left(A\right)-P(A\cap B)\ge 0$

$P\left(B\right)-P(A\cap B)\ge 0$

$\Rightarrow P\left(A\right)-P(A\cap B)+p\left(B\right)-P(A\cap B)\ge 0$ ..... (2)

From (1) and (2), we have

$P\left(A\right)-P(A\cap B)$ and $P\left(B\right)-P(A\cap B)=0$

$\Rightarrow P\left(A\right)=P(A\cap B)$ and $P\left(B\right)=P(A\cap B)$

$P\left(A\right)=P\left(B\right)$

Hence, the correct answer is option (a).