MCQ
Choose the correct answer.
If $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})$ for any two events A and B, then:
- A$\text{P(A)}=\text{P(B)}$
- B$\text{P(A)}>\text{P(B)}$
- C$\text{P}(\text{A})<\text{P(B)}$
- Dnone of these.
Choose the correct answer.
If $\text{P}(\text{A}\cup\text{B})=\text{P}(\text{A}\cap\text{B})$ for any two events A and B, then:
Solution:
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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