Applied Statistics and Probability for Engineers

2-1 SAMPLE SPACES AND EVENTS 23

complements to form other events of interest. Some of the basic set operations are summa-

rized below in terms of events:

The union of two events is the event that consists of all outcomes that are contained

in either of the two events. We denote the union as.

The intersectionof two events is the event that consists of all outcomes that are

contained in both of the two events. We denote the intersection as.

The complementof an event in a sample space is the set of outcomes in the sample

space that are not in the event. We denote the component of the event Eas.

EXAMPLE 2-6 Consider the sample space S{yy, yn, ny,nn} in Example 2-2. Suppose that the set of all out-

comes for which at least one part conforms is denoted as E 1. Then,

The event in which both parts do not conform, denoted as E 2 , contains only the single out-

come, E 2 {nn}. Other examples of events are , the null set, and E 4 S, the sample

space. If E 5 {yn, ny, nn},

EXAMPLE 2-7 Measurements of the time needed to complete a chemical reaction might be modeled with the

sample space SR, the set of positive real numbers. Let

Then,

Also,

EXAMPLE 2-8 Samples of polycarbonate plastic are analyzed for scratch and shock resistance. The results

from 50 samples are summarized as follows:

shock resistance

high low

scratch resistance high^404

low 1 5

Let Adenote the event that a sample has high shock resistance, and let Bdenote the event that a

sample has high scratch resistance. Determine the number of samples in

The event consists of the 40 samples for which scratch and shock resistances

are high. The event consists of the 9 samples in which the shock resistance is low. The

event consists of the 45 samples in which the shock resistance, scratch resistance,

or both are high.

A ́B

A¿

A ̈B

A ̈B, A¿, and A ́B.

E 1 ¿^5 x^0 x^106 and^ E 1 ¿ ̈E 2 ^5 x^0 10 x^1186

E 1 ́E 2  5 x 0 1 x 1186 and E 1 ̈E 2  5 x 0 3 x 106

E 1 ^5 x^0 1 x^106 and^ E 2 ^5 x^0 3 x^1186

E 1 ́E 5 S E 1 ̈E 5  5 yn, ny 6 E¿ 1  5 nn 6

E 3    

E 1  5 yy, yn, ny 6

E¿

E 1 ̈E 2

E 1 ́E 2

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