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.

Statement[edit]

The law of total probability is^[1] a theorem that states 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:

P(A)=\sum _{n}P(A\cap B_{n})

or, alternatively,^[1]

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

where, for any $n$ for which $P(B_{n})=0$ these terms are simply omitted from the summation, because $P(A\mid B_{n})$ is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, $P(A)$ , 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.

P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n}\mid C)

Taking the $B_{n}$ as above, and assuming $C$ is an event independent of any of the $B_{n}$ :

P(A\mid C)=\sum _{n}P(A\mid C\cap B_{n})P(B_{n})

Informal formulation[edit]

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 $P(A)$ .

Example[edit]

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}P(A)&=P(A\mid B_{X})\cdot P(B_{X})+P(A\mid B_{Y})\cdot P(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

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

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.

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