Conditional probability

Given two events A and B from the sigma-field of a probability space, with the unconditional probability of B being greater than zero (i.e., P(B) > 0), the conditional probability of A given B (${\displaystyle P(A\mid B)}$) is the probability of A occurring if B has or is assumed to have happened.[5] 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 of the probability of the joint intersection of events A and B (${\displaystyle P(A\cap B)}$)—the probability at which A and B occur together, although not necessarily occurring at the same time—and the probability of B:[2][6][7]

${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}}}$.

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 ${\displaystyle A\cap B}$ to the set B. Note that the above equation is a definition, not just a theoretical result. We denote the quantity ${\displaystyle {\frac {P(A\cap B)}{P(B)}}}$ as ${\displaystyle P(A\mid B)}$ and call it the "conditional probability of A given B."

Some authors, such as de Finetti, prefer to introduce conditional probability as an axiom of probability:

${\displaystyle P(A\cap B)=P(A\mid B)P(B)}$.

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, 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 ${\displaystyle A\cap B}$ and introduces a symmetry with the summation axiom for mutually exclusive events:[8]

${\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}$

Thus the equations can be combined to find a new representation of the :

${\displaystyle P(A\cap B)=P(A)+P(B)-P(A\cup B)=P(A\mid B)P(B)}$

${\displaystyle P(A\cup B)={P(A)+P(B)-P(A\mid B){P(B)}}}$

Conditional probability can be defined as the probability of a conditional event ${\displaystyle A_{B}}$. The Goodman–Nguyen–Van Fraassen conditional event can be defined as:

${\displaystyle A_{B}=\bigcup _{i\geq 1}\left(\bigcap _{j<i}{\overline {B}}_{j},A_{i}B_{i}\right)}$, where ${\displaystyle A_{i}}$ and ${\displaystyle B_{i}}$ represent states or elements of A or B. [9]

It can be shown that

${\displaystyle P(A_{B})={\frac {P(A\cap B)}{P(B)}}}$

which meets the Kolmogorov definition of conditional probability.[10]

If ${\displaystyle P(B)=0}$, then according to the definition, ${\displaystyle P(A\mid B)}$ is undefined.

The case of greatest interest is that of a random variable Y, conditioned on a continuous random variable X resulting in a particular outcome x. The event ${\displaystyle B=\{X=x\}}$ has probability zero and, as such, cannot be conditioned on.

Instead of conditioning on X being exactly x, we could condition on it being closer than distance ${\displaystyle \epsilon }$ away from x. The event ${\displaystyle B=\{x-\epsilon <X<x+\epsilon \}}$ will generally have nonzero probability and hence, can be conditioned on. We can then take the limit

${\displaystyle \lim _{\epsilon \to 0}P(A\mid x-\epsilon <X<x+\epsilon ).}$

For example, if two continuous random variables X and Y have a joint density ${\displaystyle f_{X,Y}(x,y)}$, then by L'Hôpital's rule and Leibniz integral rule, upon differentiation with respect to ${\displaystyle \epsilon }$:

${\displaystyle {\begin{aligned}\lim _{\epsilon \to 0}P(Y\in U\mid x_{0}-\epsilon <X<x_{0}+\epsilon )&=\lim _{\epsilon \to 0}{\frac {\int _{x_{0}-\epsilon }^{x_{0}+\epsilon }\int _{U}f_{X,Y}(x,y)\mathrm {d} y\mathrm {d} x}{\int _{x_{0}-\epsilon }^{x_{0}+\epsilon }\int _{\mathbb {R} }f_{X,Y}(x,y)\mathrm {d} y\mathrm {d} x}}\\&={\frac {\int _{U}f_{X,Y}(x_{0},y)\mathrm {d} y}{\int _{\mathbb {R} }f_{X,Y}(x_{0},y)\mathrm {d} y}}.\end{aligned}}}$

The resulting limit is the conditional probability distribution of Y given X and exists when the denominator, the probability density ${\displaystyle f_{X}(x_{0})}$, is strictly positive.

It is tempting to define the undefined probability ${\displaystyle P(A\mid X=x)}$ using this limit, but this cannot be done in a consistent manner. In particular, it is possible to find random variables X and W and values x, w such that the events ${\displaystyle \{X=x\}}$ and ${\displaystyle \{W=w\}}$ are identical but the resulting limits are not:[11]

${\displaystyle \lim _{\epsilon \to 0}P(A\mid x-\epsilon \leq X\leq x+\epsilon )\neq \lim _{\epsilon \to 0}P(A\mid w-\epsilon \leq W\leq w+\epsilon ).}$

The Borel–Kolmogorov paradox demonstrates this with a geometrical argument.

Let X be a discrete random variable and its possible outcomes denoted V. For example, if X represents the value of a rolled die then V is the set ${\displaystyle \{1,2,3,4,5,6\}}$. Let us assume for the sake of presentation that X is a discrete random variable, so that each value in V has a nonzero probability.

For a value x in V and an event A, the conditional probability is given by ${\displaystyle P(A\mid X=x)}$. Writing

${\displaystyle c(x,A)=P(A\mid X=x)}$

for short, we see that it is a function of two variables, x and A.

For a fixed A, we can form the random variable ${\displaystyle Y=c(X,A)}$. It represents an outcome of ${\displaystyle P(A\mid X=x)}$ whenever a value x of X is observed.

The conditional probability of A given X can thus be treated as a random variable Y with outcomes in the interval ${\displaystyle [0,1]}$. From the law of total probability, its expected value is equal to the unconditional probability of A.

The partial conditional probability ${\displaystyle P(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})}$ is about the probability of event ${\displaystyle A}$ given that each of the condition events ${\displaystyle B_{i}}$ has occurred to a degree ${\displaystyle b_{i}}$ (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 ${\displaystyle n}$.[12] Such ${\displaystyle n}$-bounded partial conditional probability can be defined as the conditionally expected average occurrence of event ${\displaystyle A}$ in testbeds of length ${\displaystyle n}$ that adhere to all of the probability specifications ${\displaystyle B_{i}\equiv b_{i}}$, i.e.:

${\displaystyle P^{n}(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})=\operatorname {E} ({\overline {A}}^{n}\mid {\overline {B}}_{1}^{n}=b_{1},\ldots ,{\overline {B}}_{m}^{n}=b_{m})}$[12]

Based on that, partial conditional probability can be defined as

${\displaystyle P(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})=\lim _{n\to \infty }P^{n}(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m}),}$

where ${\displaystyle b_{i}n\in \mathbb {N} }$[12]

Jeffrey conditionalization[13][14] is a special case of partial conditional probability, in which the condition events must form a partition:

${\displaystyle P(A\mid B_{1}\equiv b_{1},\ldots ,B_{m}\equiv b_{m})=\sum _{i=1}^{m}b_{i}P(A\mid B_{i})}$

When Morse code 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: ${\displaystyle P(dot\ sent\mid dot\ received)=P(dot\ received\mid dot\ sent){\frac {P(dot\ sent)}{P(dot\ received)}}.}$ 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 ${\displaystyle P(dot\ sent)={\frac {3}{7}}\ and\ P(dash\ sent)={\frac {4}{7}}}$. 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 ${\displaystyle P(dot\ received)}$.

${\displaystyle P(dot\ received)=P(dot\ received\ \cap \ dot\ sent)+P(dot\ received\ \cap \ dash\ sent)}$

${\displaystyle P(dot\ received)=P(dot\ received\mid dot\ sent)P(dot\ sent)+P(dot\ received\mid dash\ sent)P(dash\ sent)}$

${\displaystyle P(dot\ received)={\frac {9}{10}}\times {\frac {3}{7}}+{\frac {1}{10}}\times {\frac {4}{7}}={\frac {31}{70}}}$

Now, ${\displaystyle P(dot\ sent\mid dot\ received)}$can be calculated:

${\displaystyle P(dot\ sent\mid dot\ received)=P(dot\ received\mid dot\ sent){\frac {P(dot\ sent)}{P(dot\ received)}}={\frac {9}{10}}\times {\frac {\frac {3}{7}}{\frac {31}{70}}}={\frac {27}{31}}}$[16]

A geometric visualisation of Bayes' theorem. In the table, the values 3, 1, 2 and 6 give the relative weights of each corresponding condition and case. The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = P(B|A) P(A)/P(B) . Similar reasoning can be used to show that P(Ā|B) = P(B|Ā) P(Ā)/P(B) etc.

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.[19] The relationship between P(A|B) and P(B|A) is given by Bayes' theorem:

${\displaystyle {\begin{aligned}P(B\mid A)&={\frac {P(A\mid B)P(B)}{P(A)}}\\\Leftrightarrow {\frac {P(B\mid A)}{P(A\mid B)}}&={\frac {P(B)}{P(A)}}\end{aligned}}}$

That is, P(A|B) ≈ P(B|A) only if P(B)/P(A) ≈ 1, or equivalently, P(A) ≈ P(B).

In general, it cannot be assumed that P(A) ≈ P(A|B). These probabilities are linked through the law of total probability:

${\displaystyle P(A)=\sum _{n}P(A\cap B_{n})=\sum _{n}P(A\mid B_{n})P(B_{n}).}$

where the events ${\displaystyle (B_{n})}$ form a countable partition of ${\displaystyle \Omega }$.

This fallacy may arise through selection bias.[20] For example, in the context of a medical claim, let S_C 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_C) 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_C) is high. The actual probability observed by the doctor is P(S_C|H).

Not taking prior probability into account partially or completely is called base rate neglect. The reverse, insufficient adjustment from the prior probability is conservatism.