Law of total probability
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.
Part of a series on statistics |
Probability theory |
---|
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.
Notes
- 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.
- 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.
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.