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Then the unconditional probability that <math>X=1</math> is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that <math>X=1</math> conditional on <math>Y=1</math> is 1/3 (since there are three possible [[prime number]] rolls—2, 3, and 5—of which one is even).
==Conditional continuous distributions==
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==Relation to independence==
Random variables <math>X</math>, <math>Y</math> are [[Statistical independence|independent]] [[if and only if]] the conditional distribution of <math>Y</math> given <math>X</math> is, for all possible realizations of <math>X</math>, equal to the unconditional distribution of <math>Y</math>. For discrete random variables this means <math>P(Y=y|X=x) = P(Y=y)</math> for all possible <math>y</math> and <math>x</math> with <math>P(X=x)>0</math>. For continuous random variables <math>X</math> and <math>Y</math>, having a [[joint density function]], it means <math>f_Y(y|X=x) = f_Y(y)</math> for all possible <math>y</math> and <math>x</math> with <math>f_X(x)>0</math>.
==Properties==
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Also recall that an event <math>B</math> is independent of a sub-sigma field <math>\mathcal{A}</math> if <math>\mathbb{P}(B | A) = \mathbb{P}(B)</math> for all <math>A \in \mathcal{A}</math>. It is incorrect to conclude in general that the information in <math>\mathcal{A}</math> does not tell us anything about the probability of event <math>B</math> occurring. This can be shown with a counter-example:
Consider a probability space on the [[unit interval]], <math>\Omega = [0, 1]</math>. Let <math>\mathcal{G}</math> be the sigma-field of all countable sets and sets whose complement is countable. So each set in <math>\mathcal{G}</math> has measure <math>0</math> or <math>1</math> and so is independent of each event in <math>\mathcal{F}</math>. However, notice that <math>\mathcal{G}</math> also contains all the singleton events in <math>\mathcal{F}</math> (those sets which contain only a single <math>\omega \in \Omega</math>). So knowing which of the events in <math>\mathcal{G}</math> occurred is equivalent to knowing exactly which <math>\omega \in \Omega</math> occurred! So in one sense, <math>\mathcal{G}</math> contains no information about <math>\mathcal{F}</math> (it is independent of it), and in another sense it contains all the information in <math>\mathcal{F}</math>.{{sfnp|Billingsley|2012}}{{Page needed|date=May 2025}}
==See also==
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