In
probability theory
Probability theory or probability calculus 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 expre ...
, conditional probability is a measure of the
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
of an
event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred.
This particular method relies on event A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as
or occasionally . This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred):
.
For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone sick is coughing might be 75%, in which case we would have that = 5% and = 75 %. Although there is a relationship between and in this example, such a relationship or dependence between and is not necessary, nor do they have to occur simultaneously.
may or may not be equal to , i.e., the unconditional probability or absolute probability of . If , then events and are said to be
''independent'': in such a case, knowledge about either event does not alter the likelihood of each other. (the conditional probability of given ) typically differs from . For example, if a person has
dengue fever
Dengue fever is a mosquito-borne disease caused by dengue virus, prevalent in tropical and subtropical areas. Asymptomatic infections are uncommon, mild cases happen frequently; if symptoms appear, they typically begin 3 to 14 days after i ...
, the person might have a 90% chance of being tested as positive for the disease. In this case, what is being measured is that if event (''having dengue'') has occurred, the probability of (''tested as positive'') given that occurred is 90%, simply writing = 90%. Alternatively, if a person is tested as positive for dengue fever, they may have only a 15% chance of actually having this rare disease due to high
false positive
A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resu ...
rates. In this case, the probability of the event (''having dengue'') given that the event (''testing positive'') has occurred is 15% or = 15%. It should be apparent now that falsely equating the two probabilities can lead to various errors of reasoning, which is commonly seen through
base rate fallacies.
While conditional probabilities can provide extremely useful information, limited information is often supplied or at hand. Therefore, it can be useful to reverse or convert a conditional probability using
Bayes' theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ...
:
. Another option is to display conditional probabilities in a
conditional probability table
In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to display conditional probabilities of a single variable with respect to the others (i.e., the probability of each ...
to illuminate the relationship between events.
Definition
Conditioning on an event
Kolmogorov definition
Given two
events and from the
sigma-field of a probability space, with the
unconditional probability of being greater than zero (i.e., , the conditional probability of given (
) is the probability of ''A'' occurring if ''B'' has or is assumed to have happened.
''A'' is assumed to be the set of all possible outcomes of an experiment or random trial that has a restricted or reduced sample space. The conditional probability can be found by the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of the probability of the joint intersection of events and , that is,
, the probability at which ''A'' and ''B'' occur together, and the
probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
of :
[
:.
For a sample space consisting of equal likelihood outcomes, the probability of the event ''A'' is understood as the fraction of the number of outcomes in ''A'' to the number of all outcomes in the sample space. Then, this equation is understood as the fraction of the set to the set ''B''. Note that the above equation is a definition, not just a theoretical result. We denote the quantity as and call it the "conditional probability of given ."
]
As an axiom of probability
Some authors, such as de Finetti, prefer to introduce conditional probability as an axiom of probability:
:.
This equation for a conditional probability, although mathematically equivalent, may be intuitively easier to understand. It can be interpreted as "the probability of ''B'' occurring multiplied by the probability of ''A'' occurring, provided that ''B'' has occurred, is equal to the probability of the ''A'' and ''B'' occurrences together, although not necessarily occurring at the same time". Additionally, this may be preferred philosophically; under major probability interpretations
The word "probability" has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly on ...
, such as the subjective theory, conditional probability is considered a primitive entity. Moreover, this "multiplication rule" can be practically useful in computing the probability of and introduces a symmetry with the summation axiom for Poincaré Formula:
:
:Thus the equations can be combined to find a new representation of the :
:
:
As the probability of a conditional event
Conditional probability can be defined as the probability of a conditional event . The Goodman–Nguyen–Van Fraassen conditional event can be defined as:
:, where and represent states or elements of ''A'' or ''B.''
It can be shown that
:
which meets the Kolmogorov definition of conditional probability.
Conditioning on an event of probability zero
If , then according to the definition, is undefined.
The case of greatest interest is that of a random variable , conditioned on a continuous random variable resulting in a particular outcome . The event has probability zero and, as such, cannot be conditioned on.
Instead of conditioning on being ''exactly'' , we could condition on it being closer than distance away from . The event will generally have nonzero probability and hence, can be conditioned on.
We can then take the limit
For example, if two continuous random variables and have a joint density , then by L'Hôpital's rule and Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Wilhelm Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integrands ...
, upon differentiation with respect to :
:
The resulting limit is the conditional probability distribution
In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables X ...
of given and exists when the denominator, the probability density , is strictly positive.
It is tempting to ''define'' the undefined probability using limit (), but this cannot be done in a consistent manner. In particular, it is possible to find random variables and and values , such that the events and are identical but the resulting limits are not:
:
The Borel–Kolmogorov paradox demonstrates this with a geometrical argument.
Conditioning on a discrete random variable
Let be a discrete random variable and its possible outcomes denoted . For example, if represents the value of a rolled dice then is the set . Let us assume for the sake of presentation that is a discrete random variable, so that each value in has a nonzero probability.
For a value in and an event , the conditional probability
is given by .
Writing
:
for short, we see that it is a function of two variables, and .
For a fixed , we can form the random variable . It represents an outcome of whenever a value of is observed.
The conditional probability of given can thus be treated as a random variable with outcomes in the interval law of total probability
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 ev ...
, its expected value is equal to the unconditional probability
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
of .
Partial conditional probability
The partial conditional probability
is about the probability of event
given that each of the condition events
has occurred to a degree
(degree of belief, degree of experience) that might be different from 100%. Frequentistically, partial conditional probability makes sense, if the conditions are tested in experiment repetitions of appropriate length
. Such
-bounded partial conditional probability can be defined as the conditionally expected average occurrence of event
in testbeds of length
that adhere to all of the probability specifications
, i.e.:
:[
Based on that, partial conditional probability can be defined as
:
where ][
Jeffrey conditionalization
is a special case of partial conditional probability, in which the condition events must form a partition:
:
]
Example
Suppose that somebody secretly rolls two fair six-sided dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...
, and we wish to compute the probability that the face-up value of the first one is 2, given the information that their sum is no greater than 5.
* Let ''D''1 be the value rolled on dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...
1.
* Let ''D''2 be the value rolled on dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...
2.
''Probability that'' ''D''1 = 2
Table 1 shows the 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 ...
of 36 combinations of rolled values of the two dice, each of which occurs with probability 1/36, with the numbers displayed in the red and dark gray cells being ''D''1 + ''D''2.
''D''1 = 2 in exactly 6 of the 36 outcomes; thus ''P''(''D''1 = 2) = = :
:
''Probability that'' ''D''1 + ''D''2 ≤ 5
Table 2 shows that ''D''1 + ''D''2 ≤ 5 for exactly 10 of the 36 outcomes, thus ''P''(''D''1 + ''D''2 ≤ 5) = :
:
''Probability that'' ''D''1 = 2 ''given that'' ''D''1 + ''D''2 ≤ 5
Table 3 shows that for 3 of these 10 outcomes, ''D''1 = 2.
Thus, the conditional probability P(''D''1 = 2 , ''D''1+''D''2 ≤ 5) = = 0.3:
:
Here, in the earlier notation for the definition of conditional probability, the conditioning event ''B'' is that ''D''1 + ''D''2 ≤ 5, and the event ''A'' is ''D''1 = 2. We have as seen in the table.
Use in inference
In statistical inference
Statistical inference is the process of using data analysis to infer properties of an underlying probability distribution.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properties of ...
, the conditional probability is an update of the probability of an event based on new information. The new information can be incorporated as follows:[
* Let ''A'', the event of interest, be in the ]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 ...
, say (''X'',''P'').
* The occurrence of the event ''A'' knowing that event ''B'' has or will have occurred, means the occurrence of ''A'' as it is restricted to ''B'', i.e. .
* Without the knowledge of the occurrence of ''B'', the information about the occurrence of ''A'' would simply be ''P''(''A'')
* The probability of ''A'' knowing that event ''B'' has or will have occurred, will be the probability of relative to ''P''(''B''), the probability that ''B'' has occurred.
* This results in whenever ''P''(''B'') > 0 and 0 otherwise.
This approach results in a probability measure that is consistent with the original probability measure and satisfies all the Kolmogorov axioms. This conditional probability measure also could have resulted by assuming that the relative magnitude of the probability of ''A'' with respect to ''X'' will be preserved with respect to ''B'' (cf. a Formal Derivation below).
The wording "evidence" or "information" is generally used in the Bayesian interpretation of probability. The conditioning event is interpreted as evidence for the conditioned event. That is, ''P''(''A'') is the probability of ''A'' before accounting for evidence ''E'', and ''P''(''A'', ''E'') is the probability of ''A'' after having accounted for evidence ''E'' or after having updated ''P''(''A''). This is consistent with the frequentist interpretation, which is the first definition given above.
Example
When Morse code
Morse code is a telecommunications method which Character encoding, encodes Written language, text characters as standardized sequences of two different signal durations, called ''dots'' and ''dashes'', or ''dits'' and ''dahs''. Morse code i ...
is transmitted, there is a certain probability that the "dot" or "dash" that was received is erroneous. This is often taken as interference in the transmission of a message. Therefore, it is important to consider when sending a "dot", for example, the probability that a "dot" was received. This is represented by: In Morse code, the ratio of dots to dashes is 3:4 at the point of sending, so the probability of a "dot" and "dash" are . If it is assumed that the probability that a dot is transmitted as a dash is 1/10, and that the probability that a dash is transmitted as a dot is likewise 1/10, then Bayes's rule can be used to calculate .
Now, can be calculated:
Statistical independence
Events ''A'' and ''B'' are defined to be statistically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two event (probability theory), events are independent, statistically independent, or stochastically independent if, informally s ...
if the probability of the intersection of A and B is equal to the product of the probabilities of A and B:
:
If ''P''(''B'') is not zero, then this is equivalent to the statement that
:
Similarly, if ''P''(''A'') is not zero, then
:
is also equivalent. Although the derived forms may seem more intuitive, they are not the preferred definition as the conditional probabilities may be undefined, and the preferred definition is symmetrical in ''A'' and ''B''. Independence does not refer to a disjoint event.
It should also be noted that given the independent event pair Band an event C, the pair is defined to be conditionally independent if the product holds true:
This theorem could be useful in applications where multiple independent events are being observed.
Independent events vs. mutually exclusive events
The concepts of mutually independent events and mutually exclusive events
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tail ...
are separate and distinct. The following table contrasts results for the two cases (provided that the probability of the conditioning event is not zero).
In fact, mutually exclusive events cannot be statistically independent (unless both of them are impossible), since knowing that one occurs gives information about the other (in particular, that the latter will certainly not occur).
Common fallacies
:''These fallacies should not be confused with Robert K. Shope's 197
"conditional fallacy"
which deals with counterfactual examples that beg the question.''
Assuming conditional probability is of similar size to its inverse
In general, it cannot be assumed that ''P''(''A'', ''B'') ≈ ''P''(''B'', ''A''). This can be an insidious error, even for those who are highly conversant with statistics. The relationship between ''P''(''A'', ''B'') and ''P''(''B'', ''A'') is given by Bayes' theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ...
:
:
That is, ''P''(''A'', ''B'') ≈ ''P''(''B'', ''A'') only if ''P''(''B'')/''P''(''A'') ≈ 1, or equivalently, ''P''(''A'') ≈ ''P''(''B'').
Assuming marginal and conditional probabilities are of similar size
In general, it cannot be assumed that ''P''(''A'') ≈ ''P''(''A'', ''B''). These probabilities are linked through the law of total probability
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 ev ...
:
:
where the events form a countable partition of .
This fallacy may arise through selection bias
Selection bias is the bias introduced by the selection of individuals, groups, or data for analysis in such a way that proper randomization is not achieved, thereby failing to ensure that the sample obtained is representative of the population inte ...
. For example, in the context of a medical claim, let ''S'' be the event that a sequela (chronic disease) ''S'' occurs as a consequence of circumstance (acute condition) ''C''. Let ''H'' be the event that an individual seeks medical help. Suppose that in most cases, ''C'' does not cause ''S'' (so that ''P''(''S'') is low). Suppose also that medical attention is only sought if ''S'' has occurred due to ''C''. From experience of patients, a doctor may therefore erroneously conclude that ''P''(''S'') is high. The actual probability observed by the doctor is ''P''(''S'', ''H'').
Over- or under-weighting priors
Not taking prior probability into account partially or completely is called '' base rate neglect''. The reverse, insufficient adjustment from the prior probability is ''conservatism
Conservatism is a Philosophy of culture, cultural, Social philosophy, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, Convention (norm), customs, and Value (ethics and social science ...
''.
Formal derivation
Formally, ''P''(''A'' , ''B'') is defined as the probability of ''A'' according to a new probability function on the sample space, such that outcomes not in ''B'' have probability 0 and that it is consistent with all original probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s.[George Casella and Roger L. Berger (1990), ''Statistical Inference'', Duxbury Press, (p. 18 ''et seq.'')][Grinstead and Snell's Introduction to Probability](_blank)
p. 134
Let Ω be a discrete 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 ...
with elementary events , and let ''P'' be the probability measure with respect to the σ-algebra
In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
of Ω. Suppose we are told that the event ''B'' ⊆ Ω has occurred. A new probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
(denoted by the conditional notation) is to be assigned on to reflect this. All events that are not in ''B'' will have null probability in the new distribution. For events in ''B'', two conditions must be met: the probability of ''B'' is one and the relative magnitudes of the probabilities must be preserved. The former is required by the axioms of probability
The standard probability axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-wor ...
, and the latter stems from the fact that the new probability measure has to be the analog of ''P'' in which the probability of ''B'' is one - and every event that is not in ''B'', therefore, has a null probability. Hence, for some scale factor ''α'', the new distribution must satisfy:
#
#
#
Substituting 1 and 2 into 3 to select ''α'':
:
So the new probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
is
#
#
Now for a general event ''A'',
:
See also
* Bayes' theorem
Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ...
* Bayesian epistemology
* Borel–Kolmogorov paradox
* Chain rule (probability)
* Class membership probabilities
* Conditional independence
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probabi ...
* Conditional probability distribution
In probability theory and statistics, the conditional probability distribution is a probability distribution that describes the probability of an outcome given the occurrence of a particular event. Given two jointly distributed random variables X ...
* Conditioning (probability)
* Disintegration theorem
* Joint probability distribution
A joint or articulation (or articular surface) is the connection made between bones, ossicles, or other hard structures in the body which link an animal's skeletal system into a functional whole.Saladin, Ken. Anatomy & Physiology. 7th ed. McGraw- ...
* Monty Hall problem
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, based nominally on the American television game show ''Let's Make a Deal'' and named after its original host, Monty Hall. The problem was originally posed (and solved ...
* Pairwise independent distribution
* Posterior probability
The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posteri ...
* Postselection
* Regular conditional probability
References
External links
*
Visual explanation of conditional probability
{{Authority control
Mathematical fallacies
Statistical ratios