Page 40 - Applied Statistics And Probability For Engineers

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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 E ´ E . 2
1
The intersection of two events is the event that consists of all outcomes that are
contained in both of the two events. We denote the intersection as E ¨ E 2 .
1
The complement of 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 E as .
E¿
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,

E 5yy, yn, ny6
1
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 E , the null set, and E 4 S, the sample
3
space. If E 5 {yn, ny, nn},

E ´ E S E ¨ E 5yn, ny6 E¿ 5nn6
1
1
5
5
1
EXAMPLE 2-7 Measurements of the time needed to complete a chemical reaction might be modeled with the

sample space S R , the set of positive real numbers. Let
E 5x 0 1 x 106 and E 5x 0 3 x 1186
2
1
Then,

´ E 5x 0 1 x 1186 and E ¨ E 5x 0 3 x 106
E 1 2 1 2
Also,

E ¿ 5x 0 x 106 and E ¿¨ E 5x 0 10 x 1186
1
2
1
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
high 40 4
scratch resistance
low 1 5

Let A denote the event that a sample has high shock resistance, and let B denote the event that a
sample has high scratch resistance. Determine the number of samples in A ¨ B, A¿, and A ´ B.
The event A ¨ B consists of the 40 samples for which scratch and shock resistances
are high. The event A¿ consists of the 9 samples in which the shock resistance is low. The
event A ´ B consists of the 45 samples in which the shock resistance, scratch resistance,
or both are high.