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Bayes'Theorem
77
~ri
3.78 Two firms
V
and
W
consider bidding on a road
Iho
building job, which mayor may not be awarded de
leet
pending on the amounts of the bids. Firm
V
submits
ba
the
Irk
lity
3
a bid and the probability is 
that it will get the job
4
3
provided firm
W
does not bid. The probability is
4
that
W
will bid, and if it does, the probability that
V
will
1
the
get the job is only 3
luc
ring
(a) What is the probability that
V
will get the job?
(b)
If
V
gets the job, what is the probability that
W
did not bid?
W%
rom
cars
ires,
with
3.79 Engineers in charge of maintaining our nuclear fleet
must continually check for corrosion inside the pipes
that are part of the cooling systems. The inside con
dition of the pipes cannot be observed directly but a
nondestructive test can give an indication of possible
,ility
from
corrosion. This test is not infallible. The test has prob
ability 0.7 ofdetecting corrosion when it is present but
it also has probability 0.2 of falsely indicating internal
corrosion. Suppose the probability that any section of
3.16
pipe has internal corrosion is 0.1.
(a) Determine the probability that a section of pipe
has internal corrosion. given that the test indicates
its presence.
(b) Determine the probability that a section ofpipe has
internal corrosion, given that the test is negative.
3.80
An East Coast manufacturer of printed circuit boards
exposes all finished boards to an online automated ver
ification test. During one period, 900 boards were com
pleted and 890 passed the test. The test is not infallible.
Of 30 boards intentionally made to have noticeable de
fects, 25 were detected by the test. Use the relative fre
quencies to approximate the conditional probabilities
needed below.
(a)
Give an approximate value for P[Pass test
I
board
has defects].
(b) Explain why your answer in part a may
be
too
small.
(c) Give an approximate value for the probability that
a manufactured board will have defects. In order to
answer the question, you need information about
the conditional probability that a good board will
fail the test. This is important to know but was not
available at the time an answer was required. To
proceed, you can assume that this probability is
zero.
(d) Approximate the probability that a board has de
fects given that it passed the automated test.
Id the
ns
ac
prob
initial
ir
was
.
Do's and Don'ts
Do's
I.
Begin by creating a sample space
S
which specifies all possible outcomes.
2.
Always assign probabilities to events that satisfy the axioms ofprobability.
In the discrete case, the possible outcomes can be arranged in a sequence.
The axioms are then automatically satisfied when probability
Pi
is assigned
to the
ith
outcome, where
O".S
Pi
and
Pi
=
1
all outcomes in S
and the probability of any event
A
is defined as
peA) =
Pi
all outcomes in
A
3.
Combine the probabilities of events according to rules of probability.
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 Spring '08
 Mendell
 Normal Distribution, Probability, Probability distribution, Probability theory, Binomial distribution

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