# 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
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