In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the law (or formula) of total probability is a fundamental rule relating
marginal probabilities to
conditional probabilities
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occur ...
. It expresses the total probability of an outcome which can be realized via several distinct
events
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of ev ...
, hence the name.
Statement
The law of total probability is
[Zwillinger, D., Kokoska, S. (2000) ''CRC Standard Probability and Statistics Tables and Formulae'', CRC Press. page 31.] a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
that states, in its discrete case, if
is a finite or
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of a
sample space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
(in other words, a set of
pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
event
Event may refer to:
Gatherings of people
* Ceremony, an event of ritual significance, performed on a special occasion
* Convention (meeting), a gathering of individuals engaged in some common interest
* Event management, the organization of e ...
s whose
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
is the entire sample space) and each event
is
measurable
In mathematics, the concept of a measure is a generalization and formalization of Geometry#Length, area, and volume, geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly ...
, then for any event
of the same
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
:
:
or, alternatively,
[
:
where, for any for which these terms are simply omitted from the summation, because is finite.
The summation can be interpreted as a ]weighted average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
, and consequently the marginal probability, , 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 as above, and assuming is an event independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
of any of the :
:
Continuous case
The law of total probability extends to the case of conditioning on events generated by continuous random variables. Let be a probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
. Suppose is a random variable with distribution function , and an event on . Then the law of total probability states
If admits a density function , then the result is
Moreover, for the specific case where , where is a Borel set, then this yields
Example
Suppose that two factories supply light bulb
An electric light, lamp, or light bulb is an electrical component that produces light. It is the most common form of artificial lighting. Lamps usually have a base made of ceramic, metal, glass, or plastic, which secures the lamp in the soc ...
s 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:
:
where
* is the probability that the purchased bulb was manufactured by factory ''X'';
* is the probability that the purchased bulb was manufactured by factory ''Y'';
* is the probability that a bulb manufactured by ''X'' will work for over 5000 hours;
* 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 variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s. 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[''Probability: An Introduction'', by ]Geoffrey Grimmett
Geoffrey Richard Grimmett (born 20 December 1950) is a mathematician known for his work on the mathematics of random systems arising in probability theory and statistical mechanics, especially percolation theory and the contact process. He is ...
and Dominic Welsh
James Anthony Dominic Welsh (known professionally as D.J.A. Welsh) (born 29 August 1938)[law of total expectation
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...](_blank)
.
See also
* Law of total expectation
The proposition in probability theory known as the law of total expectation, the law of iterated expectations (LIE), Adam's law, the tower rule, and the smoothing theorem, among other names, states that if X is a random variable whose expected v ...
* Law of total variance In probability theory, the law of total variance or variance decomposition formula or conditional variance formulas or law of iterated variances also known as Eve's law, states that if X and Y are random variables on the same probability space, and ...
* Law of total covariance In probability theory, the law of total covariance, covariance decomposition formula, or conditional covariance formula states that if ''X'', ''Y'', and ''Z'' are random variables on the same probability space, and the covariance of ''X'' and ''Y'' ...
* Law of total cumulance
In probability theory and mathematical statistics, the law of total cumulance is a generalization to cumulants of the law of total probability, the law of total expectation, and the law of total variance. It has applications in the analysis of t ...
* Marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
Notes
References
* ''Introduction to Probability and Statistics'' by 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 Probability, Second Edition'', by John J. Schiller, Seymour Lipschutz, McGraw–Hill Professional, 2010, page 89.
* ''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.
{{DEFAULTSORT:Law Of Total Probability
Probability theorems
Statistical laws