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2-7 BAYES’ THEOREM 51

chosen every several minutes. Assume that the samples are other devices are functional. What is the probability that the
independent. circuit operates?
(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
0.9 0.9 0.8
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 0.95 0.95 0.9
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 2-92. An optical storage device uses an error recovery proce-
or not other devices are functional. What is the probability that dure that requires an immediate satisfactory readback of any
the circuit operates? 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-
0.9 0.8 0.7 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-
0.95 0.95 0.95
tains 5 that are defective. Two are selected, at random, without
replacement, from the batch. Let A and B denote the events
that the first and second container selected is defective, re-
2-91. The following circuit operates if and only if there is a spectively.
path of functional devices from left to right. The probability (a) Are A and B independent events?
each device functions is as shown. Assume that the probabil- (b) If the sampling were done with replacement, would A and
ity that a device functions does not depend on whether or not B be independent?

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,

P1A ¨ B2 P1A ƒ B2P1B2 P1B ¨ A2 P1B ƒ A2P1A2

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

P1B ƒ A2P1A2
P1Aƒ B2 for P1B2 0 (2-11)
P1B2

This is a useful result that enables us to solve for P1A ƒ B2 in terms of P1B ƒ A2.