If P(A )=P(A ) for any two events A and B, then: - Vidyadip

MCQ

If $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})$ for any two events $A$ and $B,$ then:

✓
$\text{P(A)}=\text{P(B)}$
B
$\text{P(A)}>\text{P(B)}$
C
$\text{P}(\text{A})<\text{P(B)}$
D
none of these.

✓

Answer

Correct option: A.

$\text{P(A)}=\text{P(B)}$

Given that, $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})$
$\Rightarrow\text{P}(\text{A})+\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})=\text{P}(\text{A}\cap\text{B})$
$\Rightarrow\big[\text{P(A)}-\text{P}(\text{A}\cap\text{B})\big]+\big[\text{P(B)}-\text{P}(\text{A}\cap\text{B})\big]=0$
But $\text{P(A)}-\text{P}(\text{A}\cap\text{B})\geq0\ ....(\text{i})$
$\big[\because\ \text{P}(\text{A}\cap\text{B})\leq\text{P(A)}\text{ or }\text{P(B)}\big]$
And $\text{P}(\text{B})-\text{P}(\text{A}\cap\text{B})\geq0\ ...(\text{ii})$
From eq. $(i)$ and $(ii)$ we get
$\text{P(A)}=\text{P(B)}$

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