where, for any , if , then these terms are simply omitted from the summation since is finite.
The summation can be interpreted as a weighted average, and consequently the marginal probability, , is sometimes called "average probability";[2] "overall probability" is sometimes used in less formal writings.[3]
The law of total probability can also be stated for conditional probabilities:
Taking the as above, and assuming is an event independent of any of the :
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let be a probability space. Suppose is a random variable with distribution function , and an event on . Then the law of total probability states
If admits a density function , then the result is
Moreover, for the specific case where , where is a Borel set, then this yields
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?
Applying the law of total probability, we have:
where
is the probability that the purchased bulb was manufactured by factory X;
is the probability that the purchased bulb was manufactured by factory Y;
is the probability that a bulb manufactured by X will work for over 5000 hours;
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.
The term law of total probability is sometimes taken to mean the law of alternatives, which is a special case of the law of total probability applying to discrete random variables.[citation needed] One author uses the terminology of the "Rule of Average Conditional Probabilities",[4] while another refers to it as the "continuous law of alternatives" in the continuous case.[5] This result is given by Grimmett and Welsh[6] as the partition theorem, a name that they also give to the related law of total expectation.