MAE108S22p2 (1)
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School
University of California, San Diego *
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Course
108
Subject
Aerospace Engineering
Date
Dec 6, 2023
Type
Pages
3
Uploaded by HighnessSandpiper3218
MAE 108: Probability and Statistical Methods for Mechanical
Engineering
Homework 2, Due 15 April 2022 to Gradescope by 10pm
1. During a chip shortage, an engineer needs to source semiconductors
for a project. Recent information shows the 30% of the time, semi-
conductors are available from supplier A, and 20% of the time they
are available from supplier B. The engineer is able to obtain the semi-
conductors from at least one of these two companies only 40% of the
time.
(a) What is the probability that both supplier A and B have semicon-
ductors?
(b) If supplier A does not have semiconductors in stock, what is the
probability that company B will have semiconductors in stock?
(c) Are the events of supplier A not having stock and supplier B having
stock statistically independent?
(d) Are the events of supplier A not having stock and supplier B having
stock mutually exclusive?
(e) Are the events of supplier A not having stock and supplier B having
stock collectively exhaustive?
2. Problem 2.11 Ang and Tang
3. Fertilizer and sewage are two sources of water pollution in streams.
Observations reveal that 8% of urban streams are contaminated by
fertilizer, and 5% are contaminated by sewage. Assume that the events
of contamination by sewage and fertilizer are statistically independent.
(a) Determine the probability that a stream chosen at random for
inspection is contaminated. A stream is contaminated if it contains
fertilizer or sewage or both.
(b) If the stream is found to be contaminated, what is the probability
that it is caused by sewage only?
4. Problem 2.19 Ang and Tang
5. Let
E
1
,
E
2
and
E
3
denote the events of excessive heat in the first,
second and third summers from this spring. Observations reveal that
during any summer, the probability of excessive heat is 25%. If, how-
ever, excessive heat occurred in the previous summer, the probability
of excessive heat in the following summer is increased to 35%, where
as if excessive heat occurred in the preceding two summers, the prob-
ability of excessive heat in the following summer drops to 10%.
(a) Determine the following conditional probabilities
P
(
E
2
|
E
1
)
, P
(
E
3
|
E
1
E
2
)
, P
(
E
3
|
E
2
)
(b) What is the probability that excessive heat will occur in at least
one of the next two summers?
(c) What is the probability that excessive heat will occur in each of
the next three summers?
(d) If the summer 1 did not experience excessive heat, what is the
probability that the summer 2 (the subsequent summer) would
not experience excessive heat?
6. Problem 2.38 Ang and Tang
7. Problem 2.52 Ang and Tang
8. Bayesian Updating.
When the IRS receives tax forms, it runs them through a code to flag
forms that need to be investigated further. The code looks for errors
in the data entered in the tax forms, for example inconsistencies with
W2 (income) forms or unrealistic deduction amounts.
Suppose the
code correctly flags 80% of all returns given that they have errors,
and it incorrectly flags 30% of error-free returns.
Further, suppose
that 35% of all tax returns have errors.
(a) A tax return is flagged by the code. What is the probability that
it does not contain errors, given that the code flagged it?
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