Applied Statistics and Probability for Engineers

44 CHAPTER 2 PROBABILITY

EXAMPLE 2-21 Consider the contamination discussion at the start of this section. Let Fdenote the event

that the product fails, and let Hdenote the event that the chip is exposed to high levels of

contamination. The requested probability is P(F), and the information provided can be rep-

resented as

From Equation 2-7,

which can be interpreted as just the weighted average of the two probabilities of failure.

The reasoning used to develop Equation 2-7 can be applied more generally. In the devel-

opment of Equation 2-7, we only used the two mutually exclusive Aand. However, the fact

that , the entire sample space, was important. In general, a collection of sets

such that is said to be exhaustive.A graphical dis-

play of partitioning an event Bamong a collection of mutually exclusive and exhaustive

events is shown in Fig. 2-15 on page 43.

E 1 , E 2 ,p, Ek E 1 ́ E 2 ́p ́EkS

A ́A¿S

A¿

P 1 F 2 0.10 1 0.20 2 0.005 1 0.80 2 0.0235

P 1 H 2 0.20 and P 1 H¿ 2 0.80

P 1 FƒH 2 0.10 and P 1 FƒH¿ 2 0.005

Assume are kmutually exclusive and exhaustive sets. Then

P 1 BƒE 12 P 1 E 12 P 1 BƒE 22 P 1 E 22 pP 1 BƒEk 2 P 1 Ek 2 (2-8)

P 1 B 2 P 1 B ̈E 12 P 1 B ̈E 22 pP 1 B ̈Ek 2

E 1 , E 2 ,p, Ek

Total Probability

Rule (multiple

events)

EXAMPLE 2-22 Continuing with the semiconductor manufacturing example, assume the following probabili-

ties for product failure subject to levels of contamination in manufacturing:

Probability of Failure Level of Contamination

0.10 High

0.01 Medium

0.001 Low

For any events Aand B,

P 1 B 2 P 1 B ̈A 2 P 1 B ̈A¿ 2 P 1 BƒA 2 P 1 A 2 P 1 BƒA¿ 2 P 1 A¿ 2 (2-7)

Total Probability

Rule (two events)

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