# Law of total probability

Template:Probability fundamentals

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

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

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

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. 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.

## Applications

One common application of the law is where the events coincide with a discrete random variable *X* taking each value in its range, i.e. is the event . It follows that the probability of an event *A* is equal to the expected value of the conditional probabilities of *A* given .{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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where Pr(*A* | *X*) is the conditional probability of *A* given the value of the random variable *X*.^{[3]} This conditional probability is a random variable in its own right, whose value depends on that of *X*. The conditional probability Pr(*A* | *X* = x) is simply a conditional probability given an event, [*X* = *x*]. It is a function of *x*, say *g*(*x*) = Pr(*A* | *X* = *x*). Then the conditional probability Pr(*A* | *X*) is *g*(*X*), hence itself a random variable. This version of the law of total probability says that the expected value of this random variable is the same as Pr(*A*).

This result can be generalized to continuous random variables (via continuous conditional density), and the expression becomes

where denotes the sigma-algebra generated by the random variable *X*.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## Other names

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}

^{[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

## References

- ↑
^{1.0}^{1.1}Zwillinger, D., Kokoska, S. (2000)*CRC Standard Probability and Statistics Tables and Formulae*, CRC Press. ISBN 1-58488-059-7 page 31. - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑
^{3.0}^{3.1}{{#invoke:citation/CS1|citation |CitationClass=book }} - ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑
*Probability: An Introduction*, by Geoffrey Grimmett and Dominic Welsh, Oxford Science Publications, 1986, Theorem 1B.

*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 Theory and Problems of Beginning Finite Mathematics*, by John J. Schiller, Seymour Lipschutz, and R. Alu Srinivasan, McGraw–Hill Professional, 2005, page 116.*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.