# Conditioning and Independence

Chapter

## Abstract

We have seen that the probability of a set as in Example 4 of §2.2. When Ω is countable and each point from (2.4.3), since the denominator above is equal to 1. In many questions we are interested in the proportional weight of one set Thus we are switching our attention from Ω to and call it the

*A*is its weighted proportion relative to the sample space Ω. When Ω is finite and all sample points have the same weight (therefore equally likely), then$$ P\left( A \right) = \frac{{\left| A \right|}}{{\left| \Omega \right|}} $$

*ω*has the weight*P*(*ω*) =*P*({*ω*}) attached to it, then$$ P\left( A \right) = \frac{{\sum\limits_{\omega \in A} {P\left( \omega \right)} }}{{\sum\limits_{\omega \in \Omega } {P\left( \omega \right)} }} $$

(5.1.1)

*A*relative to another set*S.*More accurately stated, this means the proportional weight of the part of*A*in*S*, namely the intersection*A ∩ S, or AS*, relative to*S.*The formula analogous to (5.1.1) is then$$ \frac{{\sum\limits_{\omega \in AS} {P\left( \omega \right)} }}{{\sum\limits_{\omega \in S} {P\left( \omega \right)} }} $$

(5.1.2)

*S*as a new universe, and considering a new proportion or probability with respect to it. We introduce the notation$$ P\left( {A\left| S \right.} \right) = \frac{{P\left( {AS} \right)}}{{P\left( S \right)}} $$

(5.1.3)

*conditional probability of A relative to S.*Other phrases such as*“given S*,”*“knowing S*,” or*“under the hypothesis*[*of*]*S*” may also be used to describe this relativity.## Keywords

Black Ball Conditional Probability Independent Random Variable General Random Variable Apriori Probability
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag New York Inc. 1979