Applied Statistics and Probability for Engineers

2-4 CONDITIONAL PROBABILITY 41

Sometimes a partition of the question into successive picks is an easier method to solve the

problem.

EXAMPLE 2-18 A day’s production of 850 manufactured parts contains 50 parts that do not meet customer

requirements. Two parts are selected randomly without replacement from the batch. What is

the probability that the second part is defective given that the first part is defective?

Let Adenote the event that the first part selected is defective, and let Bdenote the event

that the second part selected is defective. The probability needed can be expressed as

If the first part is defective, prior to selecting the second part, the batch contains 849

parts, of which 49 are defective, therefore

EXAMPLE 2-19 Continuing the previous example, if three parts are selected at random, what is the probability

that the first two are defective and the third is not defective? This event can be described in

shorthand notation as simply P(ddn). We have

The third term is obtained as follows. After the first two parts are selected, there are 848

remaining. Of the remaining parts, 800 are not defective. In this example, it is easy to obtain

the solution with a conditional probability for each selection.

EXERCISES FOR SECTION 2-4

P 1 ddn 2 

50

850



49

849



800

848

0.0032

P 1 BƒA 2  49
849

P 1 BƒA 2.

2-57. Disks of polycarbonate plastic from a supplier are an-

alyzed for scratch and shock resistance. The results from 100

disks are summarized as follows:

shock resistance

high low

scratch high 70 9

resistance low 16 5

Let Adenote the event that a disk has high shock resistance,

and let Bdenote the event that a disk has high scratch resist-

ance. Determine the following probabilities:

(a)P(A) (b)P(B)

(c) (d)

2-58. Samples of a cast aluminum part are classified

on the basis of surface finish (in microinches) and length

measurements. The results of 100 parts are summarized as

follows:

length

excellent good

surface excellent 80 2

finish good 10 8

P 1 AƒB 2 P 1 BƒA 2

Let Adenote the event that a sample has excellent surface fin-

ish, and let Bdenote the event that a sample has excellent

length. Determine:

(a) (b)

(c) (d)

(e) If the selected part has excellent surface finish, what is the

probability that the length is excellent?

(f) If the selected part has good length, what is the probability

that the surface finish is excellent?

2-59. The analysis of shafts for a compressor is summarized

by conformance to specifications:

roundness conforms

yes no

surface finish yes 345 5

conforms no 12 8

(a) If we know that a shaft conforms to roundness require-

ments, what is the probability that it conforms to surface

finish requirements?

(b) If we know that a shaft does not conform to roundness

requirements, what is the probability that it conforms to

surface finish requirements?

P 1 AƒB 2 P 1 BƒA 2

P 1 A 2 P 1 B 2

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