Page 59 - Applied Statistics And Probability For Engineers

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42 CHAPTER 2 PROBABILITY

2-60. The following table summarizes the analysis of samples of 40. In addition to the definitions of events A and B, let C
of galvanized steel for coating weight and surface roughness: denote the event that the third casting selected is from the
local supplier. Determine:
coating weight (a) P1A ¨ B ¨ C2
high low (b) P1A ¨ B ¨ C¿2
surface high 12 16 2-65. A batch of 500 containers for frozen orange juice con-
roughness low 88 34 tains 5 that are defective. Two are selected, at random, without
replacement from the batch.
(a) If the coating weight of a sample is high, what is the prob- (a) What is the probability that the second one selected is
ability that the surface roughness is high? defective given that the first one was defective?
(b) If the surface roughness of a sample is high, what is the (b) What is the probability that both are defective?
probability that the coating weight is high? (c) What is the probability that both are acceptable?
(c) If the surface roughness of a sample is low, what is the 2-66. Continuation of Exercise 2-65. Three containers are
probability that the coating weight is low? selected, at random, without replacement, from the batch.
2-61. Consider the data on wafer contamination and loca- (a) What is the probability that the third one selected is defec-
tion in the sputtering tool shown in Table 2-2. Assume that one tive given that the first and second one selected were
wafer is selected at random from this set. Let A denote the defective?
event that a wafer contains four or more particles, and let B (b) What is the probability that the third one selected is
denote the event that a wafer is from the center of the sputter- defective given that the first one selected was defective
ing tool. Determine: and the second one selected was okay?
(a) P1A2 (b) P1A ƒ B2 (c) What is the probability that all three are defective?
(c) P1B2 (d) P1B ƒ A2 2-67. A maintenance firm has gathered the following infor-
(e) P1A ¨ B2 (f) P1A ´ B2 mation regarding the failure mechanisms for air conditioning
2-62. A lot of 100 semiconductor chips contains 20 that are systems:
defective. Two are selected randomly, without replacement,
evidence of gas leaks
from the lot.
(a) What is the probability that the first one selected is defec- yes no
tive? evidence of yes 55 17
(b) What is the probability that the second one selected is electrical failure no 32 3
defective given that the first one was defective?
(c) What is the probability that both are defective? The units without evidence of gas leaks or electrical failure
(d) How does the answer to part (b) change if chips selected showed other types of failure. If this is a representative sample
were replaced prior to the next selection? of AC failure, find the probability
(a) That failure involves a gas leak
2-63. A lot contains 15 castings from a local supplier and 25
(b) That there is evidence of electrical failure given that there
castings from a supplier in the next state. Two castings are
was a gas leak
selected randomly, without replacement, from the lot of 40.
(c) That there is evidence of a gas leak given that there is
Let A be the event that the first casting selected is from the
evidence of electrical failure
local supplier, and let B denote the event that the second cast-
ing is selected from the local supplier. Determine: 2-68. If P1A ƒ B2 1 , must A B? Draw a Venn diagram to
(a) P1A2 (b) P1B ƒ A2 explain your answer.
(c) P1A ¨ B2 (d) P1A ´ B2 2-69. Suppose A and B are mutually exclusive events.
2-64. Continuation of Exercise 2-63. Suppose three cast- Construct a Venn diagram that contains the three events A, B,
ings are selected at random, without replacement, from the lot and C such that P1A ƒ C2 1 and P1B ƒ C2 0 ?

2-5 MULTIPLICATION AND TOTAL PROBABILITY RULES

2-5.1 Multiplication Rule

The 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
a multiplication rule for probabilities.