Applied Statistics and Probability for Engineers

2-7 BAYES’ THEOREM 51

chosen every several minutes. Assume that the samples are

independent.

(a) What is the probability that five successive samples were

all produced in cavity one of the mold?

(b) What is the probability that five successive samples were

all produced in the same cavity of the mold?

(c) What is the probability that four out of five successive

samples were produced in cavity one of the mold?

2-90. The following circuit operates if and only if there is a

path of functional devices from left to right. The probability

that each device functions is as shown. Assume that the prob-

ability that a device is functional does not depend on whether

or not other devices are functional. What is the probability that

the circuit operates?

2-91. The following circuit operates if and only if there is a

path of functional devices from left to right. The probability

each device functions is as shown. Assume that the probabil-

ity that a device functions does not depend on whether or not

other devices are functional. What is the probability that the

circuit operates?

2-92. An optical storage device uses an error recovery proce-

dure that requires an immediate satisfactory readback of any

written data. If the readback is not successful after three writing

operations, that sector of the disk is eliminated as unacceptable

for data storage. On an acceptable portion of the disk, the proba-

bility of a satisfactory readback is 0.98. Assume the readbacks

are independent. What is the probability that an acceptable por-

tion of the disk is eliminated as unacceptable for data storage?

2-93. A batch of 500 containers for frozen orange juice con-

tains 5 that are defective. Two are selected, at random, without

replacement, from the batch. Let Aand Bdenote the events

that the first and second container selected is defective, re-

spectively.

(a) Are Aand Bindependent events?

(b) If the sampling were done with replacement, would Aand

Bbe independent?

0.95

0.9

0.95

0.9

0.9

0.8

0.95

0.9

0.95

0.8

0.95

0.7

2-7 BAYES’ THEOREM

In some examples, we do not have a complete table of information such as the parts in Table

2-3. We might know one conditional probability but would like to calculate a different one. In

the semiconductor contamination problem in Example 2-22, we might ask the following: If

the semiconductor chip in the product fails, what is the probability that the chip was exposed

to high levels of contamination?

From the definition of conditional probability,

Now considering the second and last terms in the expression above, we can write

P 1 A ̈B 2 P 1 AƒB 2 P 1 B 2 P 1 B ̈A 2 P 1 BƒA 2 P 1 A 2

P 1 AƒB 2  (2-11)

P 1 BƒA 2 P 1 A 2

P 1 B 2

for P 1 B 2  0

This is a useful result that enables us to solve for P 1 AƒB 2 in terms of P 1 BƒA 2.

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