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)