Page 65 - Applied statistics and probability for engineers

Section 2-4/Conditional Probability 43

Let A and B denote the events that the irst and second part selected are defective, respectively. The probability requested
can be expressed as P(B | A). If the irst part is defective, prior to selecting the second part the batch contains 49 parts, of

which 2 are defective. Therefore,
P B A) = 2
|
(
49

Example 2-25 Continuing Example 2-24, what is the probability that the irst two parts selected are defective
and the third is not defective?
This probability can be described in shorthand notation as P(d d n ), where d and n denote parts
1 2 3
that are defective and not defective, respectively. Here
P d d n ) = P n d d P d d ) = P n d d P d | d P d ) = 47 2 ⋅ 3 = 0 0024
⋅
|
)
.
|
)
)
1 2
1
( 1 2
( 1 2 3
( 1
( 3
( 2
1 2
( 3
48 449 50

The probabilities for the irst and second selections are similar to those in the previous example. The P(n |d d ) is
3 1 2
based on the fact that after the irst 2 parts are selected, 1 defective and 47 nondefective parts remain.

When the probability is written to account for the order of the selections, it is easy to solve this question from the
deinition of conditional probabil ity. There are other ways to express the probability, such as P(d d n ) = P(d |d n )

1 2 3 2 1 3
P(d n ). However, such alternatives do not lead to conditional prob abilities that can be easily calculated.
1 3
Exercises FOR SECTION 2-4
Problem available in WileyPLUS at instructor’s discretion.
Tutoring problem available in WileyPLUS at instructor’s discretion
2-99. Disks of polycarbonate plastic from a supplier are 2-101. The analysis of results from a leaf transmutation
analyzed for scratch and shock resistance. The results from 100 experiment (turning a leaf into a petal) is summarized by type
disks are summarized as follows: of transformation completed:
Shock Resistance Total Textural
Transformation
High Low
Yes No
Scratch High 70 9
Total Color Yes 243 26
Resistance Low 16 5
Transformation No 13 18
Let A denote the event that a disk has high shock resistance,
and let B denote the event that a disk has high scratch resist- (a) If a leaf completes the color transformation, what is the
ance. Determine the following probabilities: probability that it will complete the textural transformation?
(a) P A ( ) (b) P B ( ) (b) If a leaf does not complete the textural transformation, what
(c) P A B| ( ) (d) P B A| ( ) is the probability it will complete the color transformation?
2-102. Samples of a cast aluminum part are classiied on the

2-100. Samples of skin experiencing desquamation are

basis of surface inish (in microinches) and length measurements.
analyzed for both moisture and melanin content. The results
The results of 100 parts are summarized as follows:
from 100 skin samples are as follows:
Melanin Content Length
High Low Excellent Good
Moisture High 13 7 Surface Excellent 80 2
Content Low 48 32 Finish Good 10 8
Let A denote the event that a sample has low melanin content, Let A denote the event that a sample has excellent surface

and let B denote the event that a sample has high moisture con- inish, and let B denote the event that a sample has excellent
tent. Determine the following probabilities: length. Determine: (b) P B ( )
(a) P A ( ) (b) P B ( ) (a) P A ( )
| (
| (
| (
(c) P A B) (d) P B A) (c) P A B) (d) P B A)
| (