Law of total probability

In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities.

Contents

Statement

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 X is not mutually exclusive with A or any of the B_n:

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

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. 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}_X[\Pr(A\mid X)] ,

where Pr(A|X) is the conditional probability of A given X,[3] and where EX denotes the expectation with respect to the random variable X.

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

References

  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 9780486636771. http://books.google.com/books?id=_mayRBczVRwC&pg=PA47. 
  3. ^ a b Deborah Rumsey (2006). Probability for dummies. For Dummies. p. 58. ISBN 9780471751410. http://books.google.com/books?id=Vj3NZ59ZcnoC&pg=PA58. 
  4. ^ Kenneth Baclawski (2008). Introduction to probability with R. CRC Press. p. 179. ISBN 9781420065213. http://books.google.com/books?id=Kglc9g5IPf4C&pg=PA179. 
  5. ^ Probability: An Introduction, by Geoffrey Grimmett and Dominic Welsh, Oxford Science Publications, 1986, Theorem 1B.