Editing Law of total probability (section)

==Example==
Suppose that two factories supply [[light bulb]]s to the market. Factory ''X''<nowiki>'</nowiki>s bulbs work for over 5000 hours in 99% of cases, whereas factory ''Y''<nowiki>'</nowiki>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?

Applying the law of total probability, we have:

: <math>
\begin{align}
P(A) & = P(A\mid B_X) \cdot P(B_X) + P(A\mid B_Y) \cdot P(B_Y) \\[4pt]
& = {99 \over 100} \cdot {6 \over 10} + {95 \over 100} \cdot {4 \over 10} = {{594 + 380} \over 1000} = {974 \over 1000}
\end{align}
</math>

where
* <math>P(B_X)={6 \over 10}</math> is the probability that the purchased bulb was manufactured by factory ''X'';
* <math>P(B_Y)={4 \over 10}</math> is the probability that the purchased bulb was manufactured by factory ''Y'';
* <math>P(A\mid B_X)={99 \over 100}</math> is the probability that a bulb manufactured by ''X'' will work for over 5000 hours;
* <math>P(A\mid B_Y)={95 \over 100}</math> is the probability that a bulb manufactured by ''Y'' will work for over 5000 hours.

Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.