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Law of total probability

## Summary

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 ${\displaystyle \left\{{B_{n}:n=1,2,3,\ldots }\right\}}$  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 ${\displaystyle B_{n}}$  is measurable, then for any event ${\displaystyle A}$  of the same probability space:

${\displaystyle P(A)=\sum _{n}P(A\cap B_{n})}$

or, alternatively,[1]

${\displaystyle P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),}$

where, for any ${\displaystyle n}$  for which ${\displaystyle P(B_{n})=0}$  these terms are simply omitted from the summation, because ${\displaystyle P(A\mid B_{n})}$  is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, ${\displaystyle P(A)}$ , 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.

${\displaystyle P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n}\mid C)}$

Taking the ${\displaystyle B_{n}}$  as above, and assuming ${\displaystyle C}$  is an event independent of any of the ${\displaystyle B_{n}}$ :

${\displaystyle P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n})}$

## Continuous case

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

${\displaystyle P(A)=\int _{-\infty }^{\infty }P(A|X=x)dF_{X}(x).}$

If ${\displaystyle X}$  admits a density function ${\displaystyle f_{X}}$ , then the result is

${\displaystyle P(A)=\int _{-\infty }^{\infty }P(A|X=x)f_{X}(x)dx.}$

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

${\displaystyle P(Y\in B)=\int _{-\infty }^{\infty }P(Y\in B|X=x)f_{X}(x)dx.}$

## 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:

{\displaystyle {\begin{aligned}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{aligned}}}

where

• ${\displaystyle P(B_{X})={6 \over 10}}$  is the probability that the purchased bulb was manufactured by factory X;
• ${\displaystyle P(B_{Y})={4 \over 10}}$  is the probability that the purchased bulb was manufactured by factory Y;
• ${\displaystyle P(A\mid B_{X})={99 \over 100}}$  is the probability that a bulb manufactured by X will work for over 5000 hours;
• ${\displaystyle P(A\mid B_{Y})={95 \over 100}}$  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.

## Notes

1. ^ a b Zwillinger, D., Kokoska, S. (2000) CRC Standard Probability and Statistics Tables and Formulae, CRC Press. ISBN 1-58488-059-7 page 31.
2. ^ Paul E. Pfeiffer (1978). Concepts of probability theory. Courier Dover Publications. pp. 47–48. ISBN 978-0-486-63677-1.
3. ^ Deborah Rumsey (2006). Probability for dummies. For Dummies. p. 58. ISBN 978-0-471-75141-0.
4. ^ Jim Pitman (1993). Probability. Springer. p. 41. ISBN 0-387-97974-3.
5. ^ Kenneth Baclawski (2008). Introduction to probability with R. CRC Press. p. 179. ISBN 978-1-4200-6521-3.
6. ^ 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.