Law of total probability

In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name. The law of total probability is a theorem that states, in its discrete case, if is a finite or countably infinite partition of a sample space (in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event is measurable, then for any event of the same sample space: or, alternatively, where, for any for which these terms are simply omitted from the summation, because is finite. The summation can be interpreted as a weighted average, and consequently the marginal probability, , is sometimes called "average probability"; "overall probability" is sometimes used in less formal writings. 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.