Page 58 - Applied Statistics And Probability For Engineers

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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 A denote the event that the first part selected is defective, and let B denote the event
that the second part selected is defective. The probability needed can be expressed as
P1B ƒ A2. If the first part is defective, prior to selecting the second part, the batch contains 849
parts, of which 49 are defective, therefore

P1B ƒ A2 49
849

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

50 49 800
P1ddn2 0.0032
850 849 848

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

2-57. Disks of polycarbonate plastic from a supplier are an- Let A denote the event that a sample has excellent surface fin-
alyzed for scratch and shock resistance. The results from 100 ish, and let B denote the event that a sample has excellent
disks are summarized as follows: length. Determine:
(a) P1A2 (b) P1B2
shock resistance
(c) P1A ƒ B2 (d) P1B ƒ A2
high low (e) If the selected part has excellent surface finish, what is the
scratch high 70 9 probability that the length is excellent?
resistance low 16 5 (f) If the selected part has good length, what is the probability
that the surface finish is excellent?
Let A denote the event that a disk has high shock resistance, 2-59. The analysis of shafts for a compressor is summarized
and let B denote the event that a disk has high scratch resist- by conformance to specifications:
ance. Determine the following probabilities:
(a) P(A) (b) P(B) roundness conforms
(c) P1A ƒ B2 (d) P1B ƒ A2 yes no
2-58. Samples of a cast aluminum part are classified surface finish yes 345 5
on the basis of surface finish (in microinches) and length
conforms no 12 8
measurements. The results of 100 parts are summarized as
follows: (a) If we know that a shaft conforms to roundness require-
ments, what is the probability that it conforms to surface
length
finish requirements?
excellent good (b) If we know that a shaft does not conform to roundness
surface excellent 80 2 requirements, what is the probability that it conforms to
finish good 10 8 surface finish requirements?