<Dl> <Dd> Pr (A ∣ C) = ∑ n Pr (A ∣ C ∩ B n) Pr (B n ∣ C) = ∑ n Pr (A ∣ C ∩ B n) Pr (B n) (\ displaystyle \ Pr (A \ mid C) = \ sum _ (n) \ Pr (A \ mid C \ cap B_ (n)) \ Pr (B_ (n) \ mid C) = \ sum _ (n) \ Pr (A \ mid C \ cap B_ (n)) \ Pr (B_ (n))) </Dd> </Dl> <Dd> Pr (A ∣ C) = ∑ n Pr (A ∣ C ∩ B n) Pr (B n ∣ C) = ∑ n Pr (A ∣ C ∩ B n) Pr (B n) (\ displaystyle \ Pr (A \ mid C) = \ sum _ (n) \ Pr (A \ mid C \ cap B_ (n)) \ Pr (B_ (n) \ mid C) = \ sum _ (n) \ Pr (A \ mid C \ cap B_ (n)) \ Pr (B_ (n))) </Dd> <P> The above mathematical statement might be interpreted as follows: given an outcome A (\ displaystyle A), with known conditional probabilities given any of the B n (\ displaystyle B_ (n)) events, each with a known probability itself, what is the total probability that A (\ displaystyle A) will happen? The answer to this question is given by Pr (A) (\ displaystyle \ Pr (A)). </P> <P> Suppose that two factories supply light bulbs to the market . Factory X's bulbs work for over 5000 hours in 99% of cases, whereas factory Y's bulbs work for over 5000 hours in 95% of cases . It is known that factory X supplies 60% of the total bulbs available and Y supplies 40% of the total bulbs available . What is the chance that a purchased bulb will work for longer than 5000 hours? </P>

When do you use the law of total probability
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