Section 5-3 The Addition Rules for Probability

Union

A union of two events A and B, denoted by (A$ \cup$B) or by (A or B), is simply the set of all the outcomes in both A and B.

\includegraphics{venn2.ps}

Example: If $ S=${1,2,3,4,5,6} and A={1,2,3} and B={2,4,6}, then (A$ \cup$B)={1,2,3,4,6}

Intersection

An intersection of two events A and B, denoted by (A$ \cap$B) or by (A and B), is the set of outcomes that A and B have in common.

\includegraphics{venn3.ps}

Example:If $ S=${1,2,3,4,5,6} and A={1,2,3} and B={2,4,6}, then (A$ \cap$B)={2}

Another Definition

Disjoint events are events which have nothing in common, i.e., P(A$ \cap$B)=0

\includegraphics{venn4.ps}

Addition Rule for Disjoint Events

If A and B are disjoint events, then P(A $ \cup$B)=P(A)+P(B)

The general addition rule

Previously, we found that P(A$ \cup$B)=P(A)+P(B) when A and B are disjoint. Now, we consider the case when A and B intersect. Hence, consider the following Venn diagram

\includegraphics{venn2.ps}

Note that when we overlap the two circles, we obtain P(A$ \cap$B) twice. Hence,

P(A$ \cup$B)=P(A)+P(B)-P(A$ \cap$B)



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