Applied Statistics and Probability for Engineers

30 CHAPTER 2 PROBABILITY

EXAMPLE 2-11 A visual inspection of a location on wafers from a semiconductor manufacturing process re-

sulted in the following table:

If one wafer is selected randomly from this process and the location is inspected, what is the

probability that it contains no particles? If information were available for each wafer, we could

define the sample space as the set of all wafers inspected and proceed as in the example with

diodes. However, this level of detail is not needed in this case. We can consider the sample space

to consist of the six categories that summarize the number of contamination particles on a wafer.

Then, the event that there is no particle in the inspected location on the wafer, denoted as E, can

be considered to be comprised of the single outcome, namely, E{0}.Therefore,

What is the probability that a wafer contains three or more particles in the inspected

location? Let Edenote the event that a wafer contains three or more particles in the inspected

location. Then, Econsists of the three outcomes {3, 4, 5 or more}. Therefore,

EXAMPLE 2-12 Suppose that a batch contains six parts with part numbers {a,b,c,d,e,f}. Suppose that two

parts are selected without replacement. Let Edenote the event that the part number of the first

part selected is a. Then Ecan be written as E{ab,ac,ad,ae,af}.The sample space can be

enumerated. It has 30 outcomes. If each outcome is equally likely,.

Also, if E 2 denotes the event that the second part selected is a, E 2 {ba,ca,da,ea,fa}

and with equally likely outcomes,.

2-2.2 Axioms of Probability

Now that the probability of an event has been defined, we can collect the assumptions that we

have made concerning probabilities into a set of axiomsthat the probabilities in any random

experiment must satisfy. The axioms ensure that the probabilities assigned in an experiment

can be interpreted as relative frequencies and that the assignments are consistent with our

intuitive understanding of relationships between relative frequencies. For example, if event A

is contained in event B, we should have. The axioms do not determine

probabilities;the probabilities are assigned based on our knowledge of the system under

study. However, the axioms enable us to easily calculate the probabilities of some events from

knowledge of the probabilities of other events.

P 1 A 2 P 1 B 2

P 1 E 22  5
30  1
6

P 1 E 2  5
30  1
6

P 1 E 2 0.100.050.100.25

P 1 E 2 0.4

Number of

Contamination

Particles Proportion of Wafers

0 0.40

1 0.20

2 0.15

3 0.10

4 0.05

5 or more 0.10

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