Conditioning and Independence

  • Kai Lai Chung
Part of the Undergraduate Texts in Mathematics book series (UTM)

Abstract

We have seen that the probability of a set 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
$$P(A) = \left| {\frac{A}{\Omega }} \right|$$
as in Example 4 of §2.2. When Ω is countable and each point ω has the weight P(ω) = P({ω}) attached to it, then
$$P(A) = \frac{{\sum\limits_{\omega \in A} {P(\omega )} }}{{\sum\limits_{\omega \in \Omega } {P(\omega )} }}$$
(5.1.1)
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 A relative to another set S. More accurately stated, this means the proportional weight of the part of A in S, namely the intersection AS,or AS,relative to S. The formula analogous to (5.1.1) is then
$$\frac{{\sum\limits_{\omega \in AS} {P(\omega )} }}{{\sum\limits_{\omega \in S} {P(\omega )} }}$$
(5.1.2)
.

Keywords

Black Ball Conditional Probability Independent Random Variable Conditional Proba Male Black Student 
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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Copyright information

© Springer Science+Business Media New York 1974

Authors and Affiliations

  • Kai Lai Chung
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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