Page 65 - Applied Statistics And Probability For Engineers

c02.qxd 5/22/02 11:16 M Page 48 RK UL 6 RK UL 6:Desktop Folder:TOD Y {22/5/2002} CH 1to3:

48 CHAPTER 2 PROBABILITY

It is left as a mind-expanding exercise to show that independence implies related results
such as

P1A¿¨ B¿2 P1A¿2P1B¿2.

The concept of independence is an important relationship between events and is used
throughout this text. A mutually exclusive relationship between two events is based only on
the outcomes that comprise the events. However, an independence relationship depends on the
probability model used for the random experiment. Often, independence is assumed to be part
of the random experiment that describes the physical system under study.

EXAMPLE 2-25 A day’s production of 850 manufactured parts contains 50 parts that do not meet customer
requirements. Two parts are selected at random, without replacement, from the batch. Let A
denote the event that the first part is defective, and let B denote the event that the second part
is defective.
We suspect that these two events are not independent because knowledge that the first
part is defective suggests that it is less likely that the second part selected is defective. Indeed,
P1B ƒ A2 49 849. Now, what is P(B)? Finding the unconditional P(B) is somewhat difficult
because the possible values of the first selection need to be considered:

P1B2 P1B ƒ A2P1A2 P1B ƒ A¿2P1A¿2
149 8492150 8502 150 84921800 8502
50 850

Interestingly, P(B), the unconditional probability that the second part selected is defec-
tive, without any knowledge of the first part, is the same as the probability that the first part
selected is defective. Yet, our goal is to assess independence. Because P1B ƒ A2 does not equal
P(B), the two events are not independent, as we suspected.

When considering three or more events, we can extend the definition of independence
with the following general result.

Definition
The events E , E , p , E n are independent if and only if for any subset of these
1
2
events E , E , p , E ,
i 1 i 2 i k
p p
¨ E ¨ ¨ E 2 P1E 2 P1E 2 P1E 2 (2-10)
i 2 i k i 1 i 2 i k
P1E i 1

This definition is typically used to calculate the probability that several events occur assuming
that they are independent and the individual event probabilities are known. The knowledge
that the events are independent usually comes from a fundamental understanding of the ran-
dom experiment.

EXAMPLE 2-26 Assume that the probability that a wafer contains a large particle of contamination is 0.01 and
that the wafers are independent; that is, the probability that a wafer contains a large particle is