Calculating Conditional Probabilities with Mutually Exclusive Events

[quantumnano]

How would I evaluate the following:
P(A|(B or C)) where B and C are two mutually exclusive events.

I have scoured a couple texts and the internet but have made no headway, any insight would be greatly appreciated.

QN

 

[mathman]

My guess: P(A|B or C) = P(A|B) + P(A|C) as long as B and C are mutually exclusive.

 

[statdad]

My guess: P(A|B or C) = P(A|B) + P(A|C) as long as B and C are mutually exclusive.

No. Consider drawing one card from a standard deck.

$$\begin{align*} A & = \text{ card is an Ace}\\ C & = \text{ card is a Club}\\ H & = \text{ card is a Heart} \end{align*}$$

Then $$C, H$$ are mutually exclusive. Now

$$\begin{align*} P(A \mid C) & = \frac 1 {13} \\ P(A \mid H) & = \frac 1 {13} \\ P(A \mid C) + P(A \mid H) & = \frac{2}{13} \\ P(A \mid C \cup H) & = \frac{2}{26} = \frac 1{13} \end{align*}$$

The problem is this (for general events)

$$P(A \mid B \cup C) = \frac{P(A \cap (B \cup C))}{P(B \cup C)} = \frac{P(A \cap B)}{P(B) + P(C)} + \frac{P(A \cap C)}{P(B) + P(C)} \tag{1}$$

and there is no simple general way to split (1) into a sum of different fractions where one has denominator $$P(B)$$ and the other has denominator $$P(C)$$.