Applied Statistics and Probability for Engineers

48 CHAPTER 2 PROBABILITY

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

such as

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 Bdenote 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,

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:

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 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.

P 1 BƒA 2

 (^50)  850
 (^149)  (^8492150)  8502  (^150)  (^84921800)  8502
P 1 B 2 P 1 BƒA 2 P 1 A 2 P 1 BƒA¿ 2 P 1 A¿ 2
P 1 BƒA 2  (^49) 849.
P 1 A¿ ̈B¿ 2 P 1 A¿ 2 P 1 B¿ 2.
The events E 1 , E 2 are independent if and only if for any subset of these
events
P 1 Ei 1 ̈Ei 2 ̈p ̈Eik 2 P 1 Ei 12 P 1 Ei 22 pP 1 Eik 2 (2-10)
Ei 1 , Ei 2 , p , Eik,
, p , En
Definition
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
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