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 ...
, conditional probability is a measure of the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of an
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 ...
occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred.
This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. 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:
.
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 unwell (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 (the unconditional 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, 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 resul ...
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:
. Another option is to display conditional probabilities in
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 po ...
to illuminate the relationship between events.
Definition
Conditioning on an event
Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
definition
Given two
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 ...
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 lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the probability of the joint intersection of events and (
)—the probability at which ''A'' and ''B'' occur together, although not necessarily occurring at the same time—and the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
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 one b ...
, 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
Undefined may refer to:
Mathematics
* Undefined (mathematics), with several related meanings
** Indeterminate form, in calculus
Computing
* Undefined behavior, computer code whose behavior is not specified under certain conditions
* 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
In calculus, l'Hôpital's rule or l'Hospital's rule (, , ), also known as Bernoulli's rule, is a theorem which provides a technique to evaluate limits of indeterminate forms. Application (or repeated application) of the rule often converts an i ...
and
Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form
\int_^ f(x,t)\,dt,
where -\infty < a(x), b(x) < \infty and the integral are , upon differentiation with respect to
:
:
The resulting limit is the
conditional probability distribution
In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the ...
of given and exists when the denominator, the probability density
, is strictly positive.
It is tempting to ''define'' the undefined probability
using this 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 In probability theory, the Borel–Kolmogorov paradox (sometimes known as Borel's paradox) is a paradox relating to conditional probability with respect to an event of probability zero (also known as a null set). It is named after Émile Borel and ...
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 die 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