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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:

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

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,

P1E2 0.4

What is the probability that a wafer contains three or more particles in the inspected
location? Let E denote the event that a wafer contains three or more particles in the inspected
location. Then, E consists of the three outcomes {3, 4, 5 or more}. Therefore,

P1E2 0.10 0.05 0.10 0.25

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 E denote the event that the part number of the first
part selected is a. Then E can 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, P1E2 5
30 1
6 .
Also, if E denotes the event that the second part selected is a, E {ba, ca, da, ea, fa}
2
2
and with equally likely outcomes, P1E 2 5
30 1
6 .
2
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 axioms that 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 P1A2 P1B2 . 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.