Introduction to Probability and Statistics for Engineers and Scientists

78 Chapter 3:Elements of Probability

Thus ifEis independent ofF, then the probability ofE’s occurrence is unchanged by
information as to whether or notFhas occurred.
Suppose now thatEis independent ofFand is also independent ofG.IsEthen
necessarily independent ofFG? The answer, somewhat surprisingly, is no. Consider the
following example.

EXAMPLE 3.8c Two fair dice are thrown. LetE 7 denote the event that the sum of the dice
is 7. LetFdenote the event that the first die equals 4 and letTbe the event that the
second die equals 3. Now it can be shown (see Problem 36) thatE 7 is independent of
Fand thatE 7 is also independent ofT; but clearlyE 7 is not independent ofFT[since
P(E 7 |FT)=1]. ■

It would appear to follow from the foregoing example that an appropriate definition
of the independence of three eventsE,F, andGwould have to go further than merely
assuming that all of the

( 3

2

)
pairs of events are independent. We are thus led to the following
definition.

Definition

The three eventsE,F, andGare said to be independent if

P(EFG)=P(E)P(F)P(G)

P(EF)=P(E)P(F)

P(EG)=P(E)P(G)

P(FG)=P(F)P(G)

It should be noted that if the eventsE, F, Gare independent, thenEwill be independent
of any event formed fromFandG. For instance,Eis independent ofF∪Gsince

P(E(F∪G))=P(EF∪EG)

=P(EF)+P(EG)−P(EFG)

=P(E)P(F)+P(E)P(G)−P(E)P(FG)

=P(E)[P(F)+P(G)−P(FG)]

=P(E)P(F∪G)

Of course we may also extend the definition of independence to more than three
events. The events E 1 ,E 2 ,...,En are said to be independent if for every subset
E 1 ′,E 2 ′,...,Er′,r≤n, of these events

P(E 1 ′E 2 ′···Er′)=P(E 1 ′)P(E 2 ′)···P(Er′)