# Law of total probability

__: Concept in probability theory__

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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

The law of total probability is^{[1]} a theorem that states, in its discrete case, if [math]\displaystyle{ \left\{{B_n : n = 1, 2, 3, \ldots}\right\} }[/math] 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 [math]\displaystyle{ B_n }[/math] is measurable, then for any event [math]\displaystyle{ A }[/math] of the same sample space:

- [math]\displaystyle{ P(A)=\sum_n P(A\cap B_n) }[/math]

or, alternatively,^{[1]}

- [math]\displaystyle{ P(A)=\sum_n P(A\mid B_n)P(B_n), }[/math]

where, for any [math]\displaystyle{ n }[/math], if [math]\displaystyle{ P(B_n) = 0 }[/math], then these terms are simply omitted from the summation since [math]\displaystyle{ P(A\mid B_n) }[/math] is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, [math]\displaystyle{ P(A) }[/math], 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:

- [math]\displaystyle{ P( {A|C} ) = \frac{{P( {A,C} )}}{{P( C )}} = \frac{{\sum\limits_n {P( {A,{B_n},C} )} }}{{P( C )}} = \frac{{\sum\limits_n P ( {A\mid {B_n},C} )P( {{B_n}\mid C} )P( C )}}{{P( C )}} = \sum\limits_n P ( {A\mid {B_n},C} )P( {{B_n}\mid C} ) }[/math]

Taking the [math]\displaystyle{ B_n }[/math] as above, and assuming [math]\displaystyle{ C }[/math] is an event independent of any of the [math]\displaystyle{ B_n }[/math]:

- [math]\displaystyle{ P(A \mid C) = \sum_n P(A \mid C,B_n) P(B_n) }[/math]

## Continuous case

The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math] be a probability space. Suppose [math]\displaystyle{ X }[/math] is a random variable with distribution function [math]\displaystyle{ F_X }[/math], and [math]\displaystyle{ A }[/math] an event on [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math]. Then the law of total probability states

[math]\displaystyle{ P(A) = \int_{-\infty}^\infty P(A |X = x) d F_X(x). }[/math]

If [math]\displaystyle{ X }[/math] admits a density function [math]\displaystyle{ f_X }[/math], then the result is

[math]\displaystyle{ P(A) = \int_{-\infty}^\infty P(A |X = x) f_X(x) dx. }[/math]

Moreover, for the specific case where [math]\displaystyle{ A = \{Y \in B \} }[/math], where [math]\displaystyle{ B }[/math] is a Borel set, then this yields

[math]\displaystyle{ P(Y \in B) = \int_{-\infty}^\infty P(Y \in B |X = x) f_X(x) dx. }[/math]

## 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 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]\displaystyle{ \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]\displaystyle{ P(B_X)={6 \over 10} }[/math] is the probability that the purchased bulb was manufactured by factory
*X*; - [math]\displaystyle{ P(B_Y)={4 \over 10} }[/math] is the probability that the purchased bulb was manufactured by factory
*Y*; - [math]\displaystyle{ 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]\displaystyle{ 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.

## 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.

^{[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.

## See also

- Law of total expectation
- Law of total variance
- Law of total covariance
- Law of total cumulance
- Marginal distribution

## Notes

- ↑
^{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. - ↑ Paul E. Pfeiffer (1978).
*Concepts of probability theory*. Courier Dover Publications. pp. 47–48. ISBN 978-0-486-63677-1. https://books.google.com/books?id=_mayRBczVRwC&pg=PA47. - ↑ Deborah Rumsey (2006).
*Probability for dummies*. For Dummies. p. 58. ISBN 978-0-471-75141-0. https://books.google.com/books?id=Vj3NZ59ZcnoC&pg=PA58. - ↑ Jim Pitman (1993).
*Probability*. Springer. p. 41. ISBN 0-387-97974-3. https://books.google.com/books?id=AoDkBwAAQBAJ&q=pitman%20probability&pg=PA41. - ↑ Kenneth Baclawski (2008).
*Introduction to probability with R*. CRC Press. p. 179. ISBN 978-1-4200-6521-3. https://books.google.com/books?id=Kglc9g5IPf4C&pg=PA179. - ↑
*Probability: An Introduction*, by Geoffrey Grimmett and Dominic Welsh, Oxford Science Publications, 1986, Theorem 1B.

## References

*Introduction to Probability and Statistics*by 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.

Original source: https://en.wikipedia.org/wiki/Law of total probability.
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