Law of total probability

Statement

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

or, alternatively,[1]

where, for any for which these terms are simply omitted from the summation, because 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 :

Informal formulation

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

Example

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

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

Notes

  1. 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. Kenneth Baclawski (2008). Introduction to probability with R. CRC Press. p. 179. ISBN 978-1-4200-6521-3.
  5. Probability: An Introduction, by Geoffrey Grimmett and Dominic Welsh, Oxford Science Publications, 1986, Theorem 1B.

References

  • 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.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.