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Milds # 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.

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• ✪ The Law of Total Probability
• ✪ Total Probability Rule
• ✪ Lesson 7 Law of Total Probability

## 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.[citation needed] 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.