Liouville's theorem (Hamiltonian)

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 the proposition that if $\left\{B_n:n=1,2,3,\dots \right\}$ 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:

$\mathrm{Pr}(A)=\sum _n\mathrm{Pr}(A\cap B_n)$

or, alternatively,

$\mathrm{Pr}(A)=\sum _n\mathrm{Pr}(A\mid B_n)\mathrm{Pr}(B_n),$

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

The summation can be interpreted as a weighted average, and consequently the marginal probability, $\mathrm{Pr}(A)$, 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 $B_n$:

$\mathrm{Pr}(A\mid C)=\sum _n\mathrm{Pr}(A\mid C\cap B_n)\mathrm{Pr}(B_n\mid C)=\sum _n\mathrm{Pr}(A\mid C\cap B_n)\mathrm{Pr}(B_n)$

Informal formulation

The above mathematical statement might be interpreted as follows: given an outcome $A$, 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 $\mathrm{Pr}(A)$.

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. What is the chance that a purchased bulb will work for longer than 5000 hours?

Applying the law of total probability, we have:

$\mathrm{Pr}(A)=\mathrm{Pr}(A|B_1)\cdot \mathrm{Pr}(B_1)+\mathrm{Pr}(A|B_2)\cdot \mathrm{Pr}(B_2)=\frac{99}{100}\cdot \frac{6}{10}+\frac{95}{100}\cdot \frac{4}{10}=\frac{594+380}{1000}=\frac{974}{1000}$

where

• $\mathrm{Pr}(B_1)=\frac{6}{10}$ is the probability that the purchased bulb was manufactured by factory X;
• $\mathrm{Pr}(B_2)=\frac{4}{10}$ is the probability that the purchased bulb was manufactured by factory Y;
• $\mathrm{Pr}(A|B_1)=\frac{99}{100}$ is the probability that a bulb manufactured by X will work for over 5000h;
• $\mathrm{Pr}(A|B_2)=\frac{95}{100}$ is the probability that a bulb manufactured by Y will work for over 5000h.

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

Applications

One common application of the law is where the events coincide with a discrete random variable X taking each value in its range, i.e. $B_n$ is the event $X=x_n$. It follows that the probability of an event A is equal to the expected value of the conditional probabilities of A given $X=x_n$.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. That is,

$\mathrm{Pr}(A)=\sum _n\mathrm{Pr}(A\mid X=x_n)\mathrm{Pr}(X=x_n)=\mathrm{E}[\mathrm{Pr}(A\mid X)],$

where Pr(A | X) is the conditional probability of A given the value of the random variable X. This conditional probability is a random variable in its own right, whose value depends on that of X. The conditional probability Pr(A | X = x) is simply a conditional probability given an event, [X = x]. It is a function of x, say g(x) = Pr(A | X = x). Then the conditional probability Pr(A | X) is g(X), hence itself a random variable. This version of the law of total probability says that the expected value of this random variable is the same as Pr(A).

This result can be generalized to continuous random variables (via continuous conditional density), and the expression becomes

$\mathrm{Pr}(A)=\mathrm{E}[\mathrm{Pr}(A\mid {\mathcal{ℱ}}_X)],$

where ${\mathcal{ℱ}}_X$ denotes the sigma-algebra generated by the random variable X.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. 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.

See also

• Law of total expectation
• Law of total variance
• Law of total cumulance

References

• Introduction to Probability and Statistics by William Mendenhall, Robert J. Beaver, Barbara M. Beaver, Thomson Brooks/Cole, 2005, page 159.
• Theory of Statistics, by Mark J. Schervish, Springer, 1995.
• Schaum's Outline of Theory and Problems of Beginning Finite Mathematics, by John J. Schiller, Seymour Lipschutz, and R. Alu Srinivasan, McGraw–Hill Professional, 2005, page 116.
• A First Course in Stochastic Models, by H. C. Tijms, John Wiley and Sons, 2003, pages 431–432.
• An Intermediate Course in Probability, by Alan Gut, Springer, 1995, pages 5–6.