condition alp rob

8/3/2019 Condition Alp Rob

1/15


2/15


3/15


4/15


5/15


6/15


7/15


8/15


9/15

Exampie 3.32 Aggregates used for highway construction are produced at threeplants with daily production volumes of 500, 1000, nd 2000 tons. Past experienceindicates that the fractions of deleterious makrials produced at th e three plantsare, respectively, 0.005, 0.008, and 0.010. If a sample of aggegate is selected a trandom from a day's total production and found ta be deleterious, which phnt islikely to have produced the sample?

AL = production from the first plant, 500 t o m per dayAZ = production from the second plant, 1000 tons per dayA3 = production rom the third plant , 2000 tons per day-

The prior probabilities are

The conditional probabilities 13, the presence of deleterious material, are

The joint occumences are

0.061Sum = E P [ A i ] IBIAiJ=-Posterior probabilities are0.005/7- 5P ( A l f B ]= - = - = 0 .08 (0.14 rior)0.06117 610.01617 16= - 0.26 (0.29 r i o r )

= 0.061/7 610.040/7 4CP[A31BJ= --- = - 0.66 (0 .57 prior)0.061/7 61


10/15

Dele~erious Probability Posterioror not of path

-- _. probabiliv r

0.061Sum = -


11/15


12/15

"+&1 &hre in &


13/15

68 BASIC PROBABILITY COiVCEPTS PRblsLEMS 69that ~(11) P(reservoir supply is high I consumption is high)- 5%Also, P (B I A) - 0% , whereas the amount ofwaterstored is independent of Ithe 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 iwater is low. What is then the probability of a water shortage?Assume that iP (AB I0 = 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 mally it does not participate in car ing the load. But if cable A breaks,Loading time T !i then B will bave to a n y he full la2 until A is replaced.(minutes) No. of observations Thc probability that A will b w k ia 0.02; also. the probability that 3i wiIl fail if it haa to carry the load by iuelf is 0.30.1 0 (a) What is the probability that bath cablei will fail?2 !5 I3 (b) If the load remains lifted. what is the probability that none of he cabIes12 i haw failed?15 2 3 2 The preliminary d e s i p o i a bridge spanning a river consists of four girdas5 I0 ! and three piers as shown in Fig. P2.22. rom considerationof the loadingand6 6 I26 resisting capacities of tach structural element the failuru probability for each0- ! girder is 10-8 and each ier is lV.Assume that failures of the girdera andBola l = 48 I piers art statistically in cpendent. Determine:! (a) Tha probability of failure in the girder(s).

(a) Sketch the histogram for the above data. (b) The pmbabiIity of failure in the pier($).(b) Based on these data, what is the probability that the loading time T ! (c) The probability of failure of the bridge system.fo r a truck wil l be at least 4 minute?(c) What is the probability that the total time for loading 2 consecutivetrucks w i l l be less than 6 minuta? Assume the loading times for an y ,two rucks to be statistically independent. I(d) In order to make a conservative estimate of the loading time, it is Iassumed that loading a truck will require at least 3 minutes; on this jassumption, what wil l be the probability that the loading time for a itruck wi l l be less than 4 minutes?

',,' 2.20 A gravity retaining wall may fail either by sliding (A ) or overturning (8 ) rboth (Fig. P2.20).Assume:(i) Probability of failureby sliding is twice as likely as that by overturning;that is , P (A ) - P (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 = tically independent.(a ) Determine he probability that sliding will occur. Ans. O.WQ9I.(b ) lf the wa l l fails, what is the probability that only sliding has occurred? (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 oneAns. 0.546. Year?2.21 Tw o cables are used to lift a load W (Fig. P2.21). However, normally only (c) If the OGcumnedof an earthquake is asrumcd in a given year, what i~cable A will be carrying the oad; cable B is slightly longer than A , so nor- the probability that the town will be flooded that year?. .-. . . -- - .___._____------- - .. .. ... , , -- . - . - -- -


14/15


15/15

condition alp rob

Documents