42 CHAPTER 2 PROBABILITY2-60. The following table summarizes the analysis of samples
of galvanized steel for coating weight and surface roughness:coating weight
high low
surface high 12 16
roughness low 88 34(a) If the coating weight of a sample is high, what is the prob-
ability that the surface roughness is high?
(b) If the surface roughness of a sample is high, what is the
probability that the coating weight is high?
(c) If the surface roughness of a sample is low, what is the
probability that the coating weight is low?
2-61. Consider the data on wafer contamination and loca-
tion in the sputtering tool shown in Table 2-2. Assume that one
wafer is selected at random from this set. Let Adenote the
event that a wafer contains four or more particles, and let B
denote the event that a wafer is from the center of the sputter-
ing tool. Determine:
(a) (b)
(c) (d)
(e) (f )
2-62. A lot of 100 semiconductor chips contains 20 that are
defective. Two are selected randomly, without replacement,
from the lot.
(a) What is the probability that the first one selected is defec-
tive?
(b) What is the probability that the second one selected is
defective given that the first one was defective?
(c) What is the probability that both are defective?
(d) How does the answer to part (b) change if chips selected
were replaced prior to the next selection?
2-63. A lot contains 15 castings from a local supplier and 25
castings from a supplier in the next state. Two castings are
selected randomly, without replacement, from the lot of 40.
Let Abe the event that the first casting selected is from the
local supplier, and let Bdenote the event that the second cast-
ing is selected from the local supplier. Determine:
(a) (b)
(c) (d)
2-64. Continuation of Exercise 2-63. Suppose three cast-
ings are selected at random, without replacement, from the lotof 40. In addition to the definitions of events Aand B, let C
denote the event that the third casting selected is from the
local supplier. Determine:
(a)
(b)
2-65. 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.
(a) What is the probability that the second one selected is
defective given that the first one was defective?
(b) What is the probability that both are defective?
(c) What is the probability that both are acceptable?
2-66. Continuation of Exercise 2-65. Three containers are
selected, at random, without replacement, from the batch.
(a) What is the probability that the third one selected is defec-
tive given that the first and second one selected were
defective?
(b) What is the probability that the third one selected is
defective given that the first one selected was defective
and the second one selected was okay?
(c) What is the probability that all three are defective?
2-67. A maintenance firm has gathered the following infor-
mation regarding the failure mechanisms for air conditioning
systems:
evidence of gas leaks
yes no
evidence of yes 55 17
electrical failure no 32 3
The units without evidence of gas leaks or electrical failure
showed other types of failure. If this is a representative sample
of AC failure, find the probability
(a) That failure involves a gas leak
(b) That there is evidence of electrical failure given that there
was a gas leak
(c) That there is evidence of a gas leak given that there is
evidence of electrical failure
2-68. If , must AB? Draw a Venn diagram to
explain your answer.
2-69. Suppose Aand Bare mutually exclusive events.
Construct a Venn diagram that contains the three events A,B,
and Csuch that P 1 AƒC 2 1 and P 1 BƒC 2 0?P 1 AƒB 2 1P 1 A ̈B ̈C¿ 2P 1 A ̈B ̈C 2P 1 A ̈B 2 P 1 A ́B 2P 1 A 2 P 1 BƒA 2P 1 A ̈B 2 P 1 A ́B 2P 1 B 2 P 1 BƒA 2P 1 A 2 P 1 AƒB 22-5 MULTIPLICATION AND TOTAL PROBABILITY RULES2-5.1 Multiplication RuleThe definition of conditional probability in Equation 2-5 can be rewritten to provide a general
expression for the probability of the intersection of two events. This formula is referred to as
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