Concept in probability theory

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]} a theorem that states, in its discrete case, if $\left\((B_{n}:n=1,2,3,\ldots }\right\))$ is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event $A$:

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

or, alternatively,^{[1]}

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

where, for any $n$, if $P(B_{n})=0$, then these terms are simply omitted from the summation since $P(A\mid B_{n})$ 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:

- $P({A|C})={\frac {P({A,C})}{P(C)))={\frac {\sum \limits _{n}{P({A,{B_{n)),C}))){P(C)))={\frac {\sum \limits _{n}P({A\mid {B_{n)),C})P(((B_{n))\mid C})P(C)}{P(C)))=\sum \limits _{n}P({A\mid {B_{n)),C})P(((B_{n))\mid C})$

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

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

##
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:

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

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

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.