Applied Statistics and Probability for Engineers

2-2 INTERPRETATIONS OF PROBABILITY 29

It is frequently necessary to assign probabilities to events that are composed of several

outcomes from the sample space. This is straightforward for a discrete sample space.

EXAMPLE 2-9 Assume that 30% of the laser diodes in a batch of 100 meet the minimum power requirements

of a specific customer. If a laser diode is selected randomly, that is, each laser diode is equally

likely to be selected, our intuitive feeling is that the probability of meeting the customer’s

requirements is 0.30.

Let Edenote the subset of 30 diodes that meet the customer’s requirements. Because

Econtains 30 outcomes and each outcome has probability 0.01, we conclude that the prob-

ability of Eis 0.3. The conclusion matches our intuition. Figure 2-10 illustrates this

example.

For a discrete sample space, the probability of an event can be defined by the reasoning

used in the example above.

For a discrete sample space, the probability of an event E,denoted as P(E), equals the

sum of the probabilities of the outcomes in E.

Definition

E

Diodes

S

P(E) = 30(0.01) = 0.30

Figure 2-10

Probability of the

event Eis the sum of

the probabilities of the

outcomes in E.

EXAMPLE 2-10 A random experiment can result in one of the outcomes {a,b,c,d} with probabilities 0.1, 0.3,

0.5, and 0.1, respectively. Let Adenote the event {a,b}, Bthe event {b,c,d}, and Cthe event

{d}.Then,

Also, , and. Furthermore, because

. Because
Because A ̈Cis the null set, P 1 A ̈C 2  0.

P 1 A ̈B 2 0.3 A ́B 5 a, b, c, d 6 , P 1 A ́B 2 0.10.30.50.11.

P 1 A¿ 2 0.6, P 1 B¿ 2 0.1 P 1 C¿ 2 0.9 A ̈B 5 b 6 ,

P 1 C 2 0.1

P 1 B 2 0.30.50.10.9

P 1 A 2 0.10.30.4

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