# Finally we give a denition of independence for any

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: if P (AB ) = P (A)P (B ), P (AC ) = P (A)P (C ) and P (BC ) = P (B )P (C ). Example 2.4.5 Suppose two fair coins are ﬂipped, so Ω = {HH, HT, T H, T T }, and the four outcomes in Ω are equally likely. Let A = {HH, HT } =“ﬁrst coin shows heads,” B = {HH, T H } =“second coin shows heads,” C = {HH, T T } =“both coins show heads or both coins show tails.” 32 CHAPTER 2. DISCRETE-TYPE RANDOM VARIABLES It is easy to check that A, B, and C are pairwise independent. Indeed, P (A) = P (B ) = P (C ) = 0.5 and P (AB ) = P (AC ) = P (BC ) = 0.25. We would consider A to be physically independent of B as well, because they involve ﬂips of diﬀerent coins. Note that P (A|BC ) = 1 = P (A). That is, knowing that both B and C are true aﬀects the probability that A is true. So A is not independent of BC. Example 2.4.5 illustrates that pairwise independence of events does not imply that any one of the events is independent of the intersection of the other two events. In order to have such independence, a stronger condition is used to deﬁne independence of three events: Deﬁnition 2.4.6 Events A, B, and C...
View Full Document

## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

Ask a homework question - tutors are online