Page 43 - Applied statistics and probability for engineers

Section 2-1/Sample Spaces and Events 21

r 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 1 ∪ E 2 .
r 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 1 ∩ E 2 .
r The complement of an event in a sample space is the set of outcomes in the sample space that
C
are not in the event. We denote the complement of the event E as E′. The notation E is also
used in other literature to denote the complement.

Example 2-6 Events Consider the sample space S = { yy yn ny nn} in Example 2-2. Suppose that the subset of
,
,
,
outcomes for which at least one camera conforms is denoted as E 1 . Then,
E 1 = { yy, yn,ny}
The event such that both cameras do not conform, denoted as E 2 , contains only the single outcome, E 2 = { nn}. Other
,
,
S
examples of events are E 3 = ∅, the null set, and E 4 = , the sample space. If E 5 = { yn ny nn},
E 1 ∪ E 5 = S E 1 ∩ E 5 = { yn,ny} E 1 = { nn}
′

Practical Interpretation: Events are used to deine outcomes of interest from a random experiment. One is often
interested in the probabilities of specii ed events.

+
Example 2-7 As in Example 2-1, camera recycle times might use the sample space S = R , the set of posi-
tive real numbers. Let
E 1 = { x 10 ≤| x 12< } and E 2 = { x 11 <| x 15< }
Then,
E 1 ∪ E 2 = { x 10 ≤ x 15}
<
|
and
E 1 ∩ E 2 = { x 11 < x 12}
<
|
Also,
E 1 ′ = { x x 10 or 12 ≤ x}
<
|
and
E 1 ′ ′ E 2 = { x 12≤ x 15}
<
|

Example 2-8 Hospital Emergency Visits The following table summarizes visits to emergency departments at
four hospitals in Arizona. People may leave without being seen by a physician, and those visits are
denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be
admitted for a stay in the hospital.
Let A denote the event that a visit is to hospital 1, and let B denote the event that the result of the visit is LWBS.
Calculate the number of outcomes in A∩ B, A , and A∪ B.
′
The event A∩ B consists of the 195 visits to hospital 1 that result in LWBS. The event A′ consists of the visits to
=
hospitals 2, 3, and 4 and contains 6991 5640+ + 4329 16 690 visits. The event A∪ B consists of the visits to hospital
,
1 or the visits that result in LWBS, or both, and contains 5292 270 246+ + = 6050 visits. Notice that the last result can
also be calculated as the number of visits in A plus the number of visits in B minus the number of visits A∩ B (that
+
would otherwise be counted twice) = 5292 953 −195 = 6050.
Practical Interpretation: Hospitals track visits that result in LWBS to understand resource needs and to improve
patient services.