# If A,B,C are three events with $P(C)=0$ then $P(A \cup B \cup C )=$

$\begin {array} {1 1} (1)\;P(A)+P(B) & \quad (2)\;P(A)+P(C) \\ (3)\;P(A)+P(B)-P(A \cap B) & \quad (4)\;None\: of \: these \end {array}$

Since $A \cap C \underline{c} C$
$P(A \cap C ) \leq P(C)=0$
Also $P(A \cap C ) \geq 0$
$\therefore P(A \cap C)=0=P(A)\: P(C)$
Similarly $P(B \cap C ) = 0$
$P ( A \cap B \cap C ) = 0$
$\therefore P ( A \cup B \cup C ) = P(A)+P(B)+P(C)-P(B \cap C )$
$-P(C \cap A)-P( A \cap B )+P(A \cap B \cap C )$
$= P(A)+P(B)-P(A \cap B)$
Ans : (3)