Page 60 - Applied statistics and probability for engineers

38 Chapter 2/Probability

Mutually
Exclusive A collection of events, E ,E , … ,E , is said to be mutually exclusive if for all pairs,
Events 1 2 k
E i ∩ E j = ∅
For a collection of mutually exclusive events,
(
P E 1) + ( ) +… ( )
P E 1 ∪ E 2 ∪…∪ ) = ( P E 2 P E k (2-8)
E k

Example 2-21 pH Here is a simple example of mutually exclusive events, which will be used quite frequently.
Let X denote the pH of a sample. Consider the event that X is greater than 6.5 but less than or equal
to 7.8. This probability is the sum of any collection of mutually exclusive events with union equal to the same range
for X. One example is
. (
. (
.
.
P 6 5 ≤
P 6 5 < X ≤ 7 8) = ( . X ≤ 7 0) + P 7 0 < X ≤ 7 5) + ( . X ≤ .
P 7 5 <
.
7 8)
Another example is
6 6) + (
. (
7 1)
7 4) + (
P 6 5 < X ≤ 7 8) = ( X ≤ . P 6 6 . < X ≤ . + ( X ≤ . P 7 4 . < X ≤ . ) 8
P 6 5 . <
P 7 1 . <
.
≤ 7
The best choice depends on the particular probabilities available.
Practical Interpretation: The partition of an event into mutually exclusive subsets is widely used in later chapters to
calculate probabilities.
Exercises FOR SECTION 2-3
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
2-82. If P A( ) = .0 3, P B( ) = .0 2 and P A( ∩ B) = .1 , deter- (b) What is the probability that a knee injury resulted from an
,
0
mine the following probabilities: activity other than a sport?
(a) P A( )′ (b) P A( ∪ B) (c) P A( ′ ′ B) 2-85. Disks of polycarbonate plastic from a supplier are
′
′
(d) P A( ∩ B ) (e) P A[( ∪ B) ] (f) P A( ′ B) analyzed for scratch and shock resistance. The results from 100
disks are summarized as follows:
2-83. If A, B, and C are mutually exclusive events with
(
P A) = .2 , P B( ) = .0 3 , and P C( ) = .0 4 , determine the follow- Shock Resistance
0
ing probabilities: High Low
(a) P A( ∪ B ∪ C) (b) P A( ∩ B ∩ C) Scratch High 70 9
(c) P A( ∩ B) (d) P A[( ∪ B) ∩ C] Resistance Low 16 5
′
(e) P A( ′ ∩ B′ ∩ C ) (a) If a disk is selected at random, what is the probability that its
2-84. In the article “ACL Reconstruction Using Bone- scratch resistance is high and its shock resistance is high?
Patellar Tendon-Bone Press-Fit Fixation: 10-Year Clinical (b) If a disk is selected at random, what is the probability that its
Results” in Knee Surgery, Sports Traumatology, Arthroscopy scratch resistance is high or its shock resistance is high?
(2005, Vol. 13, pp. 248–255), the following causes for knee (c) Consider the event that a disk has high scratch resistance and
injuries were considered: the event that a disk has high shock resistance. Are these two
events mutually exclusive?
Percentage of
Activity Knee Injuries 2-86. Strands of copper wire from a manufacturer are
analyzed for strength and conductivity. The results from 100
Contact sport 46%
strands are as follows:
Noncontact sport 44%
Activity of daily living 9% Strength
Riding motorcycle 1% High Low
(a) What is the probability that a knee injury resulted from a High conductivity 74 8
sport (contact or noncontact)? Low conductivity 15 3