Inprobability theory, thelaw (orformula)of total probability is a fundamental rule relatingmarginal probabilities toconditional probabilities. It expresses the total probability of an outcome which can be realized via several distinctevents, hence the name.

The law of total probability is^[1] atheorem that states, in its discrete case, if${\displaystyle \left\{B_n:n=1,2,3,\dots \right\}}$ is a finite orcountably infinite set ofmutually exclusive andcollectively exhaustive events, then for any event${\displaystyle A}$

$${\displaystyle P(A)=\sum _{n}P(A\cap B_n)}$$

or, alternatively,^[1]

$${\displaystyle P(A)=\sum _{n}P(A\mid B_n)P(B_n),}$$

where, for any${\displaystyle n}$, if${\displaystyle P(B_n)=0}$, then these terms are simply omitted from the summation since${\displaystyle P(A\mid B_n)}$ is finite.

The summation can be interpreted as aweighted average, and consequently the marginal probability,${\displaystyle P(A)}$, 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:

Taking the${\displaystyle B_n}$ as above, and assuming${\displaystyle C}$ is an eventindependent of any of the${\displaystyle B_n}$:

$${\displaystyle P(A\mid C)=\sum _{n}P(A\mid C,B_n)P(B_n)}$$

The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$ be aprobability space. Suppose${\displaystyle X}$ is a random variable with distribution function${\displaystyle F_X}$, and${\displaystyle A}$ an event on${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$. Then the law of total probability states

$${\displaystyle P(A)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(A|X=x)d{F}_X(x).}$$

If${\displaystyle X}$ admits a density function${\displaystyle f_X}$, then the result is

$${\displaystyle P(A)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(A|X=x)f_X(x)dx.}$$

Moreover, for the specific case where${\displaystyle A=\{Y\in B\}}$, where${\displaystyle B}$ is aBorel set, then this yields

$${\displaystyle P(Y\in B)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}P(Y\in B|X=x)f_X(x)dx.}$$

Suppose that two factories supplylight bulbs to the market. FactoryX's bulbs work for over 5000 hours in 99% of cases, whereas factoryY's bulbs work for over 5000 hours in 95% of cases. It is known that factoryX 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:

where

• ${\displaystyle P(B_X)=\frac{6}{10}}$ is the probability that the purchased bulb was manufactured by factoryX;
• ${\displaystyle P(B_Y)=\frac{4}{10}}$ is the probability that the purchased bulb was manufactured by factoryY;
• ${\displaystyle P(A\mid B_X)=\frac{99}{100}}$ is the probability that a bulb manufactured byX will work for over 5000 hours;
• ${\displaystyle P(A\mid B_Y)=\frac{95}{100}}$ is the probability that a bulb manufactured byY will work for over 5000 hours.

Thus each purchased light bulb has a 97.4% chance to work for more than 5000 hours.

The termlaw of total probability is sometimes taken to mean thelaw of alternatives, which is a special case of the law of total probability applying todiscrete 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 thepartition theorem, a name that they also give to the relatedlaw of total expectation.

• Law of large numbers
• Law of total expectation
• Law of total variance
• Law of total covariance
• Law of total cumulance
• Marginal distribution

• Theorems in probability theory
• Statistical laws

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