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Law of total probability

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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

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The law of total probability is[1] a theorem that states, in its discrete case, if is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event

or, alternatively,[1]

where, for any , if , then these terms are simply omitted from the summation since is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, , 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 as above, and assuming is an event independent of any of the :

Continuous case

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The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let be a probability space. Suppose is a random variable with distribution function , and an event on . Then the law of total probability states

If admits a density function , then the result is

Moreover, for the specific case where , where is a Borel set, then this yields

Example

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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:

where

  • is the probability that the purchased bulb was manufactured by factory X;
  • is the probability that the purchased bulb was manufactured by factory Y;
  • is the probability that a bulb manufactured by X will work for over 5000 hours;
  • 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

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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.

See also

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Notes

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  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

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  • 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.