Is $P( A \cup B\, |\, C) $ the same as $P(A | C) + P(B | C) $ ( $A$ and $B$ are mutually exclusive)

Question

Is $P( A \cup B \,|\, C)$ the same as $P(A | C) + P(B | C)$ ? Here $A$ and $B$ are mutually exclusive.

Hint: it's the same if they are conditionally mutually exclusive; and they are, if they are mutually exclusive. — leonbloy, Oct 19 '13 at 21:32

score 3 · Accepted Answer · answered Oct 19 '13 at 21:32

Yes.

Remember: $$ \begin{align*} P(A\cup B\mid C)&=\frac{P((A\cup B)\cap C)}{P(C)}\\ &=\frac{P((A\cap C)\cup (B\cap C))}{P(C)}\\ &=\frac{P(A\cap C)+P(B\cap C)}{P(C)}\\ &=\frac{P(A\cap C)}{P(C)}+\frac{P(B\cap C)}{P(C)}\\ &=P(A\mid C)+P(B\mid C). \end{align*} $$ (In the third line, we used the fact that if $A\cap B=\varnothing$, then $(A\cap C)\cap (B\cap C)=\varnothing$.)

score 3 · Answer 2 · answered Oct 19 '13 at 21:37

3

Hint: Remember,

$A\cap B= \emptyset \implies (A\cap C) \cap ( B\cap C)=\emptyset\quad \forall C$.
$(A\cup B)\cap C= (A\cap C)\cup (B\cap C)$
$P\big(A\cup B \big| C\big) = \dfrac{P\big((A\cup B)\cap C\big)}{P(C)}$

answered Oct 19 '13 at 21:37

Elias Costa

14,658

Is $P( A \cup B\, |\, C) $ the same as $P(A | C) + P(B | C) $ ( $A$ and $B$ are mutually exclusive)

2 Answers2

Linked