Page 76 - Applied statistics and probability for engineers

54 Chapter 2/Probability

a device is functional does not depend on whether or not other Let A and B denote the event that the irst bar is wide and B
devices are functional. What is the probability that the circuit denote the event that the second bar is wide. Determine the
operates? following:
(a) P(A) (b) P(B) (c) P A( ∩ B)
0.9 0.9 0.8
(d) Are A and B independent events?
2-163. An integrated circuit contains 10 million logic gates
(each can be a logical AND or OR circuit). Assume the prob-
0.95 0.95 0.9
ability of a gate failure is p and that the failures are independent.
The integrated circuit fails to function if any gate fails. Deter-
2-158. Consider the endothermic reactions in Exercise 2-50. Let
A denote the event that a reaction's inal temperature is 271 K or mine the value for p so that the probability that the integrated
less. Let B denote the event that the heat absorbed is above target. circuit functions is 0.95.
Are these events independent? 2-164. Table 2-1 provides data on wafers categorized by
2-159. Consider the hospital emergency room data in Example location and contamination levels. Let A denote the event
2-8. Let A denote the event that a visit is to hospital 4, and let B that contamination is low, and let B denote the event that
denote the event that a visit results in LWBS (at any hospital). the location is center. Are A and B independent? Why or
Are these events independent? why not?
2-160. Consider the well failure data in Exercise 2-53. Let A 2-165. Table 2-1 provides data on wafers categorized by loca-
denote the event that the geological formation has more than tion and contamination levels. More generally, let the number
1000 wells, and let B denote the event that a well failed. Are of wafers with low contamination from the center and edge
these events independent? locations be denoted as n and n , respectively. Similarly, let
le
lc
2-161. A Web ad can be designed from four different colors, n and n denote the number of wafers with high contamina-
he
hc
three font types, ive font sizes, three images, and ive text tion from the center and edge locations, respectively. Suppose
phrases. A speciic design is randomly generated by the Web that n = 10n and n = 10n . That is, there are 10 times as
he
le
hc
lc
server when you visit the site. Let A denote the event that the many low con tamination wafers as high ones from each loca-
design color is red, and let B denote the event that the font tion. Let A denote the event that contamination is low, and let
size is not the smallest one. Are A and B independent events? B denote the event that the location is center. Are A and B inde-
Explain why or why not. pendent? Does your conclusion change if the multiplier of 10
(between low and high contamination wafers) is changed from
2-162. Consider the code in Example 2-12. Suppose that all
40 codes are equally likely (none is held back as a delimiter). 10 to another positive integer?
2-7 Bayes’ Theorem
The examples in this chapter indicate that information is often presented in terms of conditional
probabilities. These conditional probabilities commonly provide the probability of an event (such
as failure) given a condition (such as high or low contamination). But after a random experiment
generates an outcome, we are naturally interested in the probability that a condition was present
(high contamination) given an outcome (a semiconductor failure). Thomas Bayes addressed this
essential question in the 1700s and developed the fundamental result known as Bayes’ theorem.
Do not let the simplicity of the mathematics conceal the importance. There is extensive interest in
such probabilities in modern statistical analysis.
From the deinition of conditional probability,
P B A P A ( )
(
P A B P B ( ) = (
P A∩ B) = ( | ) P B ∩ A) = ( | )
Now, considering the second and last terms in the preceding expression, we can write

)
(
(
P B A P A)
|
P A B) = for P B ( ) > 0 (2-15)
|
(
P B)
(
This is a useful result that enables us to solve for P A B( | ) in terms of P B A( | ).