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

The law of total probability is 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,

$\Pr(A)=\sum _{n}\Pr(A\mid B_{n})\Pr(B_{n}),$

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"; "overall probability" is sometimes used in less formal writings.

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 event $A$, 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 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:

${\begin{aligned}\Pr(A)&=\Pr(A\mid B_{X})\cdot \Pr(B_{X})+\Pr(A\mid B_{Y})\cdot \Pr(B_{Y})\\[4pt]&={99 \over 100}\cdot {6 \over 10}+{95 \over 100}\cdot {4 \over 10}={{594+380} \over 1000}={974 \over 1000}\end{aligned}}$

where

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

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. This result is given by Grimmett and Welsh as the partition theorem, a name that they also give to the related law of total expectation.

See also

• Law of total expectation
• Law of total variance
• Law of total cumulance
• Marginal distribution

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.