Question

Assertion :Let $A,B$ and $C$ be three events such that $P\left(C\right)=0$.

$P(A\cap B\cap C)=0$

Reason: $P(A\cup B\cup C)=P(A\cup B)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$A\cap B\cap C\subseteq C\Rightarrow P(A\cap B\cap C)\le P\left(C\right)=0$
$\therefore P(A\cap B\cap C)=0$
Similarly, $P(A\cap C)=0,P(B\cap C)=0$
Next, $P(A\cup B\cup C)=P(A\cup B)+P\left(C\right)-P\left[\right(A\cup B)\cap C)]$
But $P\left[\right(A\cup B)\cap C]=P\left[\right(A\cap C)\cup (B\cap C\left)\right]$
$=P(A\cap C)+P(B\cap C)-P(A\cap B\cap C)=0$
Thus, $P(A\cup B\cup C)=P(A\cup B)$