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Formally, ''P''(''A'' | ''B'') is defined as the probability of ''A'' according to a new probability function on the sample space, such that outcomes not in ''B'' have probability 0 and that it is consistent with all original [[probability measure]]s.<ref>George Casella and Roger L. Berger (1990), ''Statistical Inference'', Duxbury Press, {{ISBN|0-534-11958-1}} (p. 18 ''et seq.'')</ref><ref name="grinstead">[http://math.dartmouth.edu/~prob/prob/prob.pdf Grinstead and Snell's Introduction to Probability], p. 134</ref>
Let Ω be a discrete [[sample space]] with [[elementary event]]s {''ω''}, and let ''P'' be the probability measure with respect to the [[σ-algebra]] of Ω. Suppose we are told that the event ''B'' ⊆ Ω has occurred. A new probability distribution (denoted by the conditional notation) is to be assigned on {''ω''} to reflect this. All events that are not in ''B'' will have null probability in the new distribution. For events in ''B'', two conditions must be met: the probability of ''B'' is one and the relative magnitudes of the probabilities must be preserved. The former is required by the [[Probability axioms|axioms of probability]], and the latter stems from the fact that the new probability measure has to be the analog of ''P'' in which the probability of ''B'' is one - and every event that is not in ''B'', therefore, has a null probability. Hence, for some scale factor ''α'', the new distribution must satisfy:
#<math>\omega \in B : P(\omega\mid B) = \alpha P(\omega)</math>
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