Law of total probability

Probability theory

Probability axioms
Probability space Sample space Elementary event Event Random variable Probability measure
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
Venn diagram Tree diagram

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

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 $C$ is an event independent with any of the $B_n$ :

\Pr(A \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n \mid C) = \sum_n \Pr(A \mid C \cap B_n) \Pr(B_n)

Informal formulation

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

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. What is the chance that a purchased bulb will work for longer than 5000 hours?

Applying the law of total probability, we have:

${\Pr(A)=\Pr(A|B_1)}\cdot{\Pr(B_1)}+{\Pr(A|B_2)}\cdot{\Pr(B_2)}={99 \over 100}\cdot{6 \over 10}+{95 \over 100}\cdot{4 \over 10}={{594 + 380} \over 1000}={974 \over 1000}$

where

$\Pr(B_1)={6 \over 10}$ is the probability that the purchased bulb was manufactured by factory X;
$\Pr(B_2)={4 \over 10}$ is the probability that the purchased bulb was manufactured by factory Y;
$\Pr(A|B_1)={99 \over 100}$ is the probability that a bulb manufactured by X will work for over 5000 hours;
$\Pr(A|B_2)={95 \over 100}$ 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.

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

where Pr(A | X) is the conditional probability of A given the value of the random variable X.^[3] This conditional probability is a random variable in its own right, whose value depends on that of X. The conditional probability Pr(A | X = x) is simply a conditional probability given an event, [X = x]. It is a function of x, say g(x) = Pr(A | X = x). Then the conditional probability Pr(A | X) is g(X), hence itself a random variable. This version of the law of total probability says that the expected value of this random variable is the same as Pr(A).

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.

References

↑ 1.0 1.1 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.
↑ 3.0 3.1 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.

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.
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.