In probability theory, the law of total probability is that "the prior probability of A is equal to the prior expected value of the posterior probability of A." That is, for any random variable N,

\Pr(A)=E[\Pr(A\mid N)]

where \scriptstyle{\Pr(A\mid N)} is the conditional probability of A given N.

Law of alternatives

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. It is the proposition that if { B_n : n = 1, 2, 3, ... } is a finite or countably infinite partition of a probability space and each set B_n is measurable, then for any event A we have

\Pr(A)=\sum_{n} \Pr(A\cap B_n)\,

or, alternatively,

\Pr(A)=\sum_{n} \Pr(A\mid B_n)\Pr(B_n).\,

See also

• law of total expectation
• law of total variance
• law of total cumulance

References

• 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.
• CRC Standard Probability and Statistics Tables and Formulae, by Daniel Zwillinger and Stephen Kokoska, CRC Press, 2000, page 31.
• 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.

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