World Library  
Flag as Inappropriate
Email this Article

Law of total probability

Article Id: WHEBN0000312383
Reproduction Date:

Title: Law of total probability  
Author: World Heritage Encyclopedia
Language: English
Subject: Conditioning (probability), Conditional probability, Bayes' theorem, Probability axioms, Inductive probability
Collection: Probability Theorems, Statistical Laws
Publisher: World Heritage Encyclopedia

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.


  • Statement 1
  • Informal formulation 2
  • Example 3
  • Applications 4
  • Other names 5
  • See also 6
  • References 7


The law of total probability is[1] the proposition that if \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 B_n is measurable, then for any event A of the same probability space:

\Pr(A)=\sum_n \Pr(A\cap B_n)\,

or, alternatively,[1]

\Pr(A)=\sum_n \Pr(A\mid B_n)\Pr(B_n),\,

where, for any n\, for which \Pr(B_n) = 0 \, these terms are simply omitted from the summation, because \Pr(A\mid B_n)\, is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, \Pr(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. Taking the B_n as above, and assuming C is an event independent with any of the B_n:

\Pr(A \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n)

Informal formulation

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


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:

{\Pr(A)=\Pr(A|B_1)}\cdot{\Pr(B_1)}+{\Pr(A|B_2)}\cdot{\Pr(B_2)}={99 \over 100}\cdot{6 \over 10}+{95 \over 100}\cdot{4 \over 10}={{594 + 380} \over 1000}={974 \over 1000}


  • \Pr(B_1)={6 \over 10} is the probability that the purchased bulb was manufactured by factory X;
  • \Pr(B_2)={4 \over 10} is the probability that the purchased bulb was manufactured by factory Y;
  • \Pr(A|B_1)={99 \over 100} is the probability that a bulb manufactured by X will work for over 5000h;
  • \Pr(A|B_2)={95 \over 100} is the probability that a bulb manufactured by Y will work for over 5000h.

Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.


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. B_n is the event X=x_n. It follows that the probability of an event A is equal to the expected value of the conditional probabilities of A given X=x_n. That is,

\Pr(A)=\sum_n \Pr(A\mid X=x_n)\Pr(X=x_n) = \operatorname{E}[\Pr(A\mid X)] ,

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

\Pr(A)= \operatorname{E}[\Pr(A\mid \mathcal{F}_X)],

where \mathcal{F}_X denotes the sigma-algebra generated by the random variable X.

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. 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


  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.  
  3. ^ a b Deborah Rumsey (2006). Probability for dummies. For Dummies. p. 58.  
  4. ^ Kenneth Baclawski (2008). Introduction to probability with R. CRC Press. p. 179.  
  5. ^ 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.
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.