全概率公式

又因为 （公式略）

此处Pr(A | B)是B发生后A的条件概率，所以全概率公式又可写作： （公式略）

全概率公式将对一复杂事件A的概率求解问题转化为了在不同情况或不同原因 Bn下发生的简单事件的概率的求和问题。

条件概率的期望

在离散情况下，上述公式等于下面这个公式。但后者在连续情况下仍然成立： （公式略）

此处N是任意随机变量。

这个公式还可以表达为：

"A的先验概率等于A的后验概率的事前期望。

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 a theorem that states, in its discrete case, if （公式略） is a finite or countably infinite set of mutually exclusive and collectively exhaustive events, then for any event A

P(A)=∑ ₙP(A∩ Bₙ)

or, alternatively, P(A)=∑ ₙP(A| Bₙ)P(Bₙ),

where, for any n, if P(Bₙ)=0, then these terms are simply omitted from the summation since P(A| Bₙ) is finite.

The summation can be interpreted as a weighted average, and consequently the marginal probability, P(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: （公式略）

Taking the Bₙ as above, and assuming C is an event independent of any of the Bₙ: P(A| C)=∑ ₙP(A| C,Bₙ)P(Bₙ)

Continuous case

The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let (Ω,F,P) be a probability space. Suppose X is a random variable with distribution function F_(X), and A an event on (Ω,F,P). Then the law of total probability states P(A)=∫ _(-∞)^(∞)P(A|X=x)dF_(X)(x).

X admits a density function f_(X), then the result is P(A)=∫ _(-∞)^(∞)P(A|X=x)f_(X)(x)dx.

Moreover, for the specific case where （公式略）, where B is a Borel set, then this yields P(Y∈ B)=∫ _(-∞)^(∞)P(Y∈ B|X=x)f_(X)(x)dx.

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: （公式略）

where （公式略） is the probability that the purchased bulb was manufactured by factory X; （公式略） is the probability that the purchased bulb was manufactured by factory Y; （公式略） is the probability that a bulb manufactured by X will work for over 5000 hours; （公式略） 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 uses the terminology of the "Rule of Average Conditional Probabilities", while another refers to it as the "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.