Exampie 3.32
Aggregates
used for highway construction are produced at three
plants with daily production volumes of 500, 1000,and 2000 tons. Past experience
indicates that the fractions of deleterious makrials produced at the three plants
are, respectively, 0.005, 0.008, and 0.010. If a sample of aggegate is selected at
random from a day's total production and found ta be deleterious, which phnt is
likely to have produced the sample?
AL = production from the first plant, 500 t o m per day
AZ = production from the second plant, 1000 tons per day
A3 = production £rom the third p l a n t , 2000 tons per day-
The prior probabilities are
The conditional probabilities 13, the presence of deleterious material, are
The joint occumences are
Sum = E P [ A i ]PIBIAiJ =
0.061
7
Posterior probabilities are
0.005/7- 5
P(AlfB]= = - = 0.08
0.06117
61
0.01617 16
=
= 0.26
= 0.061/7 61
-
0.040/7
4C
-
(0.14prior)
(0.29prior)
P[A31BJ= --=
= 0.66 (0.57 prior)
0.061/7 6 1
Dele~erious Probability
or not
of path
-_.
Sum =
0.061
7
Posterior
probabiliv
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PRblsLEMS
BASIC PROBABILITY COiVCEPTS
that
69
~(11
C)E P(reservoir supply is high I consumption is high)
I
-
-
75%
I
Also, P(B A) 50%, whereas the amount ofwaterstored is independent of
the demand.
Suppose that a water shortage will occur when thew is high demand (or :
consumption) for water but either the reservoir supply is low or the stored i
water is low. What is then the probability of a water shortage? Assume that i
!
P ( A B 0 = 0.5 P ( A 3 ) .
2.19 The time T (in minutes) that it takes to load crushed rocks from a quarry !
onto a truck varies considerably. Prom a record of 48 loadings, the following [
were observed.
I
i
(minutes)
No. of observations
1
2
0
5
12
15
3
4
5
i
!
I
i
I0
6
!
I
6
0
Tolal = 48
26
mally it does not participate in car ing the load. But if cable A breaks,
then B will bave to a n y the full
until A is replaced.
Thc probability that A will b w k ia 0.02; also. the probability that 3
wiIl fail if it haa to carry the load by iuelf is 0.30.
(a) What is the probability that bath cablei will fail?
(b) If the load remains lifted. what is the probability that none of the cabIes
haw failed?
232 The preliminary d e s i p o i a bridge spanning a river consists of four girdas
and three piers as shown in Fig. P2.22.From consideration of the loading and
resisting capacities of tach structural element the failuru probability for each
girder is 10-8 and each ier is lV. Assume that failures of the girdera and
piers art statistically in cpendent. Determine:
la2
!i
Loading time T
!
B
I!
(a) Sketch the histogram for the above data.
(b) Based on these data, what is the probability that the loading time T !,
for a truck will be at least 4 minute?
(c) What is the probability that the total time for loading 2 consecutive
trucks w i l l be less than 6 minuta? Assume the loading times for any ,
I
two trucks to be statistically independent.
(d) In order to make a conservative estimate of the loading time, it is I
assumed that loading a truck will require at least 3 minutes; on this j
assumption, what will be the probability that the loading time for a i
truck will be less than 4 minutes?
',,' 2.20 A gravity retaining wall may fail either by sliding (A) or overturning (8)or
(a) Tha probability of failure in the girder(s).
(b) The pmbabiIity of failure in the pier($).
(c) The probability of failure of the bridge system.
both (Fig. P2.20). Assume:
(i) Probability of failure by sliding is twice as likely as that by overturning;
that is, P ( A )
I
-
2P(B).
( i i ) Probability that the wall also fairs by sliding, given that it has failed by
overturning, P ( A ) B) = 0.8
(iii) Probabiliry of failure of wall =
(a) Determine he probability that sliding will occur. Ans. O.WQ9I.
(b) lf the wall fails, what is the probability that only sliding has occurred?
Ans. 0.546.
2.21 Two cables are used to lift a load W (Fig. P2.21). However, normally only
cable A will be carrying the load; cable B is slightly longer than A , so nor-
-
. .-..
. .- ... .
------.___._____-------
-
,
tically independent.
(a) What I# the probability of an mrthqunlc-induced flood in a p r l
(b) What is the probability that the town in free from flooding in any one
Year?
(c) If the OGcumned of an earthquake is asrumcd in a given year, what i~
the probability that the town will be flooded that year?
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