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Unformatted text preview: Math 210 — Topics in Finite Math
Spring 2007 — Final Exam Name :
TA : VERSION A (a) The exam has 7 problems. (b) Show your work for every problem. Answers Without justiﬁcation Will not receive credit.
(0) Calculators are not ailowed. (d) There is a blank page at'the end of the exam for if you need it. i
i
i 1. (25 points) Out of every million gallons of milk produced in Wisconsin, only 3 gallons are not
pasteurized. A gallon of pasteurized milk has a 99%chance of being fresh after 5 days, While a
gallon of milk that has not been pasteurized has only a 50% chance of being fresh after 5 days. If you buy a gallon of Wisconsin milk and it is no longer fresh after 5 days, What is the chance
that it has not been pasteurized? 2. (a) (10 points) How many 5—letter ” words” can you make from the letters of the word apple? (b) (10 points) HOW many of such words start with the letter p? r 3. (30 points) Starting January 1st of 2008 Emmy will put 100 dollars in her new bank account
v every ﬁrst day of the month. Interest is added to her account on the at next—to—last day of every
month, the interest’rate is 4% per year compounded monthly. She will also put an extra 1, 000
dollars in her bank amount on the ﬁrst days of 2009 and 2010. How much money will she have
in her account on the last day of 2010? [The dates are set up slightly/differently from the way it is done in the book so be careful when using the formula for the amount of an annuity] 4. A Markov Chain is given by the following matrix: [0 1“
EM} (a) (10 points) Show that the Markov chain is regular. (b) (10 points) The initial state vector is X0 = [1,0]. Compute X4. (C) (15 points) Using the initial state vector from part (b), estimate What Xaooo is. 5. (a) (15 points) Sketch the feasible set for the following constraints:
a: 2 0, y 2 0,
a: + 2y S 12, 2113+y_>_2, Clearly mark all the corners and give their coordinates. (b) (10 points) What are the maximum and minimum of the objective function 32: + 4y? 6. (25 points) We will have to pay 5,000 dollars 5 years from today, 6,000 six years from today
and 7, 000 seven years from today. Our bank gives us a 4% interest per year compounded every
halfyear. How much money should we put in our account today if we want to have exactly
enough money to be able to pay these two amounts later? 1‘ 7. An urn contains 5 balls: 4 white ones and 1 purple ball. We repeatedly draw a ball from the um
and put it back. We do this 25 times.
(a) (10 points) What is the chance that we draw exactly 4 purple balls? (b) (10 points) Is random approximation valid? (explain your answer as always) (20 points) Use random approximation to estimate the chance that we draw at least 4 purple
balls (you may use the table on the next page). TABLE A.1 Areas under the Standard Normal Curve z i .00 .01
00 ' .0000 .0040
00 .0398 .0438
02 .0793 .0832
03  1179 .1217
0,4 .1554 .1591
05 .1915 .1950
0.8 .2257 .2291
. 0.7 .2580 .2811
0.8 .2881 .2910
09 .3159 3188 ‘
1.0 .3413 .3438
1.1 .3843 .3885
L2 .3849 .3889
1.3 .4032 .4049
1.4 .4192 .4207
1.5 .4332 .4345
1.6 .4452 .4483
1.7 .4554 .4584
1.8 .4841 .4849
1.9 .4713 .4719
2.0 .4772 .4778
21 .4821 .4828
2.2 .4881 .4884
2.3 .4893 . .4898
2.4 .4918 .4920
2.5 .4938 .4940
2.6 .4953 .4955
2.7 .4988 .4988
2.8 .4974 .4975
2.9 .4981 .4982
30 4987 .4987 .02 .0080
.0478
.0871
_.1255
.1828 ' .1985
.2324
.2642
.2939
.3212 .3461
.3686
.3888
.4066
.4222 .4357
.4474
.4573
.4656
.4726 .4783
.4830
.4888
.4898
.4922 .4941
.4956
.4967
.4976
.4982 .4987 .03 .0120
.0517
.0910
.1293
.1664 .2019
.2357
.2673
.2967
.3238 .3485
.3708
.3907
.4082
.4236 .4370
.4484
.4582
.4664
.4732 .4788
.4834 ‘.4871 .4901 4925 .4943
.4957
.4968
.4977
.4983 .4988 O4
.0160
.0557
.0948 .1331
.1700 .2054
.2389
.2704
.2995
.3264 .3508
.3729
.3925
.4099
.4251 .4382
.4495
.4591
.4671
.4738 .4793
.4838
.4875
.4904
.4927 .4945
.4959
.4989
.4977
.4984 .4988 06 .0239
.0636
.1026
.1406
.1772 .2123
.2454
.2764 .3051 .3315 .3554
.3770
.3962
.4131
.4279 .4406
.4515
.4608
.4686
.4750 .4803
.4846
.4881 .4909' .4931 .4948
.4961
.4971
.4979
.4985 .4989 07 .0279
.0875
.1084
.1443
.1808 .2157
.2486
.2794
.3078
.3340 .3577
.3790 '.3980 .4147
.4292 .4418
.4525
.4616
.4693
.4756 .4808
.4850
.4884
.4911
.4932 .4949
.4962 44972 .4979
.4985 .4989 08 .0319
.0714
.1103
.1480
'.1844 .2190
.2517
@2823
.3106
.3365 .3599
.3810 .39975 '.4162
.4306 .4429' .4535
.4625
.4699
.4761 .4812
.4854
.4887
.4913
.4934 .4951. .4963
.4973
.4980
.4986 .4990 09 .0359
.0753
.1141
.1517
.1879 .2224
.2549
.2852
.3133 .3389 .3621
.3830
.4015
.4177
.4319 .4441 .4545. .4633
.4706
.4767 .4817
.4857
.4890
.4916
.4936 .4952
.4964
.4974
.4981
.4986 .4990 W ...
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 Fall '08
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 Math, bank account, Pasteurization, initial state vector

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