I need to show by example that the following formula is not always true:
$P(A ∣ B ∪ C) = P(A ∣ B) + P(A ∣ C), B ∩ C = NULL $
2026-04-06 03:34:37.1775446477
Conditional probabilities prove false
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Let there be one blue card and two red cards. The blue card has a black dot on it. One red card has a black dot and the other a white dot on it. Pick a card at random. Define
$A$ - 'The card is red'
$B$ - 'The card has a white dot on it'
$C$ - 'The card has a black dot on it'
Compute $P(A) = 2/3$, $P(B) = 1/3$, $P(C) = 2/3$. Also $P(A\cap B) = 1/3 $ and $P(A\cap C) = 1/3$. Then $$P(A|B\cup C) = 2/3,\qquad P(A|B) = 1,\qquad P(A|C) = 1/2 $$ so the RHS wouldn't even be a probability.