Page 67 - Applied statistics and probability for engineers

Section 2-5/Multiplication and Total Probability Rules 45

2-113. Consider the endothermic reactions in Exercise 2-50. 2-117. Consider the bar code in Example 2-12. Suppose that
Let A denote the event that a reaction's i nal temperature is 271 all 40 codes are equally likely (none is held back as a delimiter).
K or less. Let B denote the event that the heat absorbed is above Determine the probability for each of the following:

target. Determine the following probabilities. (a) The second bar is wide given that the irst bar is wide.
′ (
| (
(a) P A B) (b) P A B) (b) The third bar is wide given that the irst two bars are not wide.
|

| (
(c) P A B′) (d) P B A) (c) The irst bar is wide given that the last bar is wide.
| (

2-114. Consider the hospital emergency room data in Exam- 2-118. Suppose that a patient is selected randomly from those
described in Exercise 2-57. Let A denote the event that the
ple 2-8. Let A denote the event that a visit is to hospital 4, and
patient is treated with ribavirin plus interferon alfa, and let B
let B denote the event that a visit results in LWBS (at any hos-
pital). Determine the following probabilities. denote the event that the response is complete. Determine the
(
(a) P A B| ( ) (b) P A B′ | ) following probabilities:
| (
| (
(c) P A B′) (d) P B A) (a) P B A( | ) (b) P A B( | )
(c) P A B( | ′ ) (d) P A B( ′ | )
2-115. Consider the well failure data in Exercise 2-53.
2-119. Suppose that a patient is selected randomly from those
(a) What is the probability of a failure given there are more
described in Exer cise 2-98. Let A denote the event that the patient
than 1,000 wells in a geological formation?
is in group 1, and let B denote the event that there is no progres-
(b) What is the probability of a failure given there are fewer
sion. Determine the following probabilities:
than 500 wells in a geological formation?
(a) P B A( | ) (b) P A B( | )
2-116. An article in the The Canadian Entomologist (Har- (c) P A B( | ′ (d) P A B( ′
court et al., 1977, Vol. 109, pp. 1521–1534) reported on the ) | )
2-120. A computer system uses passwords that contain
life of the alfalfa weevil from eggs to adulthood. The follow-
exactly eight characters, and each character is one of the 26
ing table shows the number of larvae that survived at each
lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10
stage of development from eggs to adults.
integers (0–9). Let Ω denote the set of all possible passwords.
Early Late Pre- Late
Eggs Adults Suppose that all passwords in Ω are equally likely. Determine
Larvae Larvae pupae Pupae the probability for each of the following:
421 412 306 45 35 31 (a) Password contains all lowercase letters given that it con-
(a) What is the probability an egg survives to adulthood? tains only letters
(b) What is the probability of survival to adulthood given (b) Password contains at least 1 uppercase letter given that it
survival to the late larvae stage? contains only letters
(c) What stage has the lowest probability of survival to the (c) Password contains only even numbers given that is con-
next stage? tains all numbers
2-5 Multiplication and Total Probability Rules
The probability of the intersection of two events is often needed. The conditional probability dei -
nition in Equation 2-9 can be rewritten to provide a formula known as the multiplication rule for
probabilities.
Multiplication
(
)
Rule P A∩ B) = ( | ) P A B P B ( ) (2-10)
P B A P A ( ) = ( |
The last expression in Equation 2-10 is obtained by interchanging A and B.
Example 2-26 Machining Stages The probability that the i rst stage of a numerically controlled machining

operation for high-rpm pistons meets speciications is 0.90. Failures are due to metal variations,
i xture alignment, cutting blade condition, vibration, and ambient environmental conditions. Given that the i rst stage
meets speciications, the probability that a second stage of machining meets speciications is 0.95. What is the prob-

ability that both stages meet specii cations?

Let A and B denote the events that the irst and second stages meet speciications, respectively. The probability requested is
(
P A∩ B) = ( | ) 0 95 . ) = .855
P B A P A ( ) = . (0
90
0
(
Although it is also true that P A∩ B) = ( | ) (
P A B P B , ) the information provided in the problem does not match this
second formulation.
Practical Interpretation: The probability that both stages meet speciications is approximately 0.85, and if additional stages

were needed to complete a piston, the probability would decrease further. Consequently, the probability that each stage is
completed successfully needs to be large in order for a piston to meet all specii cations.