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Concept in probability theoryPart of a series on statisticsProbability theoryProbabilityAxiomsDeterminismSystemIndeterminismRandomnessProbability spaceSample spaceEventCollectively exhaustive eventsElementary eventMutual exclusivityOutcomeSingletonExperimentBernoulli trialProbability distributionBernoulli distributionBinomial distributionNormal distributionProbability measureRandom variableBernoulli processContinuous or discreteExpected valueMarkov chainObserved valueRandom walkStochastic processComplementary eventJoint probabilityMarginal probabilityConditional probabilityIndependenceConditional independenceLaw of total probabilityLaw of large numbersBayes theoremBooles inequalityVenn diagramTree diagramvteIn 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.StatementThe law of total probability is1 a theorem that states, in its discrete case, if B n : n = 1 , 2 , 3 , … displaystyle leftB_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 displaystyle B_n is measurable, then for any event A displaystyle A of the same probability space: P ( A ) = ∑ n P ( A ∩ B n ) displaystyle P(A)=sum _nP(Acap B_n) or, alternatively,1 P ( A ) = ∑ n P ( A ∣ B n ) P ( B n ) , displaystyle P(A)=sum _nP(Amid B_n)P(B_n), where, for any n displaystyle n for which P ( B n ) = 0 displaystyle P(B_n)=0 these terms are simply omitted from the summation, because P ( A ∣ B n ) displaystyle P(Amid B_n) is finite.The summation can be interpreted as a weighted average, and consequently the marginal probability, P ( A ) displaystyle P(A) , is sometimes called average probability;2 overall probability is sometimes used in less formal writings.3The law of total probability can also be stated for conditional probabilities: P ( A | C ) = P ( A , C ) P ( C ) = ∑ n P ( A , B n , C ) P ( C ) = ∑ n P ( A ∣ B n , C ) P ( B n ∣ C ) P ( C ) P ( C ) = ∑ n P ( A ∣ B n , C ) P ( B n ∣ C ) displaystyle P(A|C)=frac P(A,C)P(C)=frac sum limits _nP(A,B_n,C)P(C)=frac sum limits _nP(Amid B_n,C)P(B_nmid C)P(C)P(C)=sum limits _nP(Amid B_n,C)P(B_nmid C) Taking the B n displaystyle B_n as above, and assuming C displaystyle C is an event independent of any of the B n displaystyle B_n : P ( A ∣ C ) = ∑ n P ( A ∣ C ∩ B n ) P ( B n ) displaystyle P(Amid C)=sum _nP(Amid Ccap B_n)P(B_n) Continuous caseThe law of total probability extends to the case of conditioning on events generated by continuous random variables. Let ( Ω , F , P ) displaystyle (Omega ,mathcal F,P) be a probability space. Suppose X displaystyle X is a random variable with distribution function F X displaystyle F_X , and A displaystyle A an event on ( Ω , F , P ) displaystyle (Omega ,mathcal F,P) . Then the law of total probability states P ( A ) = ∫ − ∞ ∞ P ( A | X = x ) d F X ( x ) . displaystyle P(A)=int _-infty ^infty P(A|X=x)dF_X(x). If X displaystyle X admits a density function f X displaystyle f_X , then the result is P ( A ) = ∫ − ∞ ∞ P ( A | X = x ) f X ( x ) d x . displaystyle P(A)=int _-infty ^infty P(A|X=x)f_X(x)dx. Moreover, for the specific case where A = Y ∈ B displaystyle A=Yin B , where B displaystyle B is a Borel set, then this yields P ( Y ∈ B ) = ∫ − ∞ ∞ P ( Y ∈ B | X = x ) f X ( x ) d x . displaystyle P(Yin B)=int _-infty ^infty P(Yin B|X=x)f_X(x)dx. ExampleSuppose that two factories supply light bulbs to the market. Factory Xs bulbs work for over 5000 hours in 99% of cases, whereas factory Ys 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: P ( A ) = P ( A ∣ B X ) ⋅ P ( B X ) + P ( A ∣ B Y ) ⋅ P ( B Y ) = 99 100 ⋅ 6 10 + 95 100 ⋅ 4 10 = 594 + 380 1000 = 974 1000 displaystyle beginalignedP(A)&=P(Amid B_X)cdot P(B_X)+P(Amid B_Y)cdot P(B_Y)4pt&=99 over 100cdot 6 over 10+95 over 100cdot 4 over 10=594+380 over 1000=974 over 1000endaligned where P ( B X ) = 6 10 displaystyle P(B_X)=6 over 10 is the probability that the purchased bulb was manufactured by factory X; P ( B Y ) = 4 10 displaystyle P(B_Y)=4 over 10 is the probability that the purchased bulb was manufactured by factory Y; P ( A ∣ B X ) = 99 100 displaystyle P(Amid B_X)=99 over 100 is the probability that a bulb manufactured by X will work for over 5000 hours; P ( A ∣ B Y ) = 95 100 displaystyle P(Amid 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 namesThe 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 Welsh6 as the partition theorem, a name that they also give to the related law of total expectation.See alsoLaw of total expectationLaw of total varianceLaw of total covarianceLaw of total cumulanceMarginal distributionNotes^ 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.^ Jim Pitman (1993). Probability. Springer. p. 41. ISBN 0-387-97974-3.^ 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.ReferencesIntroduction to Probability and Statistics by Robert J. Beaver, Barbara M. Beaver, Thomson BrooksCole, 2005, page 159.Theory of Statistics, by Mark J. Schervish, Springer, 1995.Schaums 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.Retrieved from https:en.wikipedia.orgwindex.php?title=Law_of_total_probability&oldid=
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