# Law of total probability

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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.

## Statement[edit]

The law of total probability is^{[1]} the proposition that 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 probability space:

or, alternatively,^{[1]}

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";^{[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 with any of the :

## Informal formulation[edit]

The above mathematical statement might be interpreted as follows: *given an outcome , with known conditional probabilities given any of the events, each with a known probability itself, what is the total probability that will happen?* The answer to this question is given by .

## Example[edit]

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.

## Other names[edit]

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 even uses the terminology "continuous law of alternatives" in the continuous case.

^{[4]}This result is given by Grimmett and Welsh

^{[5]}as the

**partition theorem**, a name that they also give to the related law of total expectation.

## See also[edit]

## Notes[edit]

- ^
^{a}^{b}Zwillinger, D., Kokoska, S. (2000)*CRC Standard Probability and Statistics Tables and Formulae*, CRC Press. ISBN 1-58488-059-7 page 31. **^**Paul E. Pfeiffer (1978).*Concepts of probability theory*. Courier Dover Publications. pp. 47–48. ISBN 978-0-486-63677-1.**^**Deborah Rumsey (2006).*Probability for dummies*. For Dummies. p. 58. ISBN 978-0-471-75141-0.**^**Kenneth Baclawski (2008).*Introduction to probability with R*. CRC Press. p. 179. ISBN 978-1-4200-6521-3.**^***Probability: An Introduction*, by Geoffrey Grimmett and Dominic Welsh, Oxford Science Publications, 1986, Theorem 1B.

## References[edit]

*Introduction to Probability and Statistics*by William Mendenhall, Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.*Theory of Statistics*, by Mark J. Schervish, Springer, 1995.*Schaum's Outline of Probability, Second Edition*, by John J. Schiller, Seymour Lipschutz, McGraw–Hill Professional, 2010, page 89.*A First Course in Stochastic Models*, by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.*An Intermediate Course in Probability*, by Alan Gut, Springer, 1995, pages 5–6.