Conditional probability

The conditional probability of an event relative to another event is a characteristic connecting the two events. If $A$ and $B$ are events and ${\mathsf P} ( B) > 0$, then the conditional probability ${\mathsf P} ( A \mid B )$ of the event $A$ relative to (or under the condition, or with respect to) $B$ is defined by the equation

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

The conditional probability ${\mathsf P} ( A \mid B )$ can be regarded as the probability that the event $A$ is realized under the condition that $B$ has taken place. For independent events $A$ and $B$ the conditional probability ${\mathsf P} ( A \mid B )$ coincides with the unconditional probability ${\mathsf P} ( A)$.

About the connection between the conditional and unconditional probabilities of events see Bayes formula and Complete probability formula.

The conditional probability of an event $A$ with respect to a $\sigma$- algebra $\mathfrak B$ is a random variable ${\mathsf P} ( A \mid \mathfrak B )$, measurable relative to $\mathfrak B$, for which

$$\int\limits _ { B } {\mathsf P} ( A \mid \mathfrak B ) {\mathsf P} ( d \omega ) = \ {\mathsf P} ( A \cap B )$$

for any $B \in \mathfrak B$. The conditional probability with respect to a $\sigma$- algebra is defined up to equivalence.

If the $\sigma$- algebra $\mathfrak B$ is generated by a countable number of disjoint events $B _ {1} , B _ {2} \dots$ having positive probability and the union of which coincides with the whole space $\Omega$, then

$${\mathsf P} ( A \mid \mathfrak B ) = \ {\mathsf P} ( A \mid B _ {k} ) \ \ \textrm{ for } \omega \in B _ {k} ,\ \ k = 1 , 2 ,\dots .$$

The conditional probability of an event $A$ with respect to the $\sigma$- algebra $\mathfrak B$ can be defined as the conditional mathematical expectation ${\mathsf E} ( I _ {A} \mid \mathfrak B )$ of the indicator function of $A$.

Let $( \Omega , {\mathcal A} , {\mathsf P} )$ be a probability space and let $\mathfrak B$ be a subalgebra of ${\mathcal A}$. The conditional probability ${\mathsf P} ( A \mid \mathfrak B )$ is called regular if there exists a function $p ( \omega , A )$, $\omega \in \Omega$, $A \in {\mathcal A}$, such that

a) for a fixed $\omega$ the function $p ( \omega , A )$ is a probability on the $\sigma$- algebra ${\mathcal A}$;

b) ${\mathsf P} ( A \mid \mathfrak B ) = p ( \omega , A )$ with probability one.

For a regular conditional probability the conditional mathematical expectation can be expressed by integrals, with the conditional probability taking the role of the measure.

The conditional probability with respect to a random variable $X$ is defined as the conditional probability with respect to the $\sigma$- algebra generated by $X$.

References

[1]	A.N. Kolmogorov, "Foundations of the theory of probability" , Chelsea, reprint (1950) (Translated from Russian)
[2]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[3]	M. Loève, "Probability theory" , Princeton Univ. Press (1963)