Applied Statistics and Probability for Engineers

If Aand Bare mutually exclusive events,

P 1 A ́ B 2 P 1 A 2 P 1 B 2 (2-2)

34 CHAPTER 2 PROBABILITY

the probability that a wafer was either at the edge or that it contains four or more particles? Let

E 1 denote the event that a wafer contains four or more particles, and let E 2 denote the event

that a wafer is at the edge.

The requested probability is. Now, and. Also,

from the table,. Therefore, using Equation 2-1, we find that

What is the probability that a wafer contains less than two particles or that it is both at the

edge and contains more than four particles? Let E 1 denote the event that a wafer contains less

than two particles, and let E 2 denote the event that a wafer is both from the edge and contains

more than four particles. The requested probability is. Now, and

. Also, E 1 and E 2 are mutually exclusive. Consequently, there are no wafers in
the intersection and. Therefore,

Recall that two events Aand Bare said to be mutually exclusive if. Then,

, and the general result for the probability of simplifies to the third ax-

iom of probability.

P 1 A ̈B 2  0 A ́B

A ̈B   

P 1 E 1 ́E 22 0.600.030.63

P 1 E 1 ̈E 22  0

P 1 E 22 0.03

P 1 E 1 ́E 22 P 1 E 12 0.60

P 1 E 1 ́E 22 0.150.28 0.040.39

P 1 E 1 ̈E 22 0.04

P 1 E 1 ́E 22 P 1 E 12 0.15 P 1 E 22 0.28

Table 2-2 Wafers Classified by Contamination and Location

Number of

Contamination

Particles Center Edge Totals

0 0.30 0.10 0.40

1 0.15 0.05 0.20

2 0.10 0.05 0.15

3 0.06 0.04 0.10

4 0.04 0.01 0.05

5 or more 0.07 0.03 0.10

Totals 0.72 0.28 1.00

Three or More Events

More complicated probabilities, such as , can be determined by repeated use

of Equation 2-1 and by using some basic set operations. For example,

P 1 A ́B ́C 2 P 31 A ́B 2 ́C 4 P 1 A ́B 2 P 1 C 2 P 31 A ́B 2 ̈C 4

P 1 A ́ B ́ C 2

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