Revision date 01/27/96.
Solutions written by J.C. Turner fall 1995.
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Corrections to these answers should be e-mailed to gof@usna.navy.mil.
4. 1/2, 5/9
8. (Assume problem should read "generates *4* successive digits")
a. 10^4
b. 1/10^4
c. 10*9*8*7/10^4
d. 10/10^4
e. 4*10*9/10^4
12. # outcomes=C(10+8+5,3)=1771
a. 186/1771
b. 400/1771
14. 0.0098, 0.0294, 0.0313 (assuming assoc. prof. are the only ones picked from for two of the
talks and the greeting is given by either a prof. or an asst. prof.
16. 30 (assuming w o r)
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4. 0.408, 0.132, 0.221
8. 0.225, 0.554, 0.75, 0.75, 0.625
12. (a) 0.1, (b) 0.26, 0.34, (c) 0.31, (d) 0.625
14. 0.135, 0.055, 0.667, 0.988
I have found the following errors in solutions of the odd numbered problems supplied to us.
4.5 a. 6.5% of the population reads exactly on magazine.
4.11 d. .4 (vice 0.367)
4.15 a. 0.03/.17=0.1765
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8.3 The following table is for N=12, p=0.2 and x from 0 to 5.
The probability of passing is 1-SUM = 0.01941.
0 0.06872 1 0.20616 2 0.28347 3 0.23622 4 0.13288 5 0.05315 0.01941
8.4 Table for N=12, p=0.7, x=0 to 5
0.9613992 = 1-SUM
0 5.314E-07 1 1.488E-05 2 0.000191 3 0.0014853 4 0.0077977 5 0.0291115 0.0386008
8.6 The following table gives the binomial probabilities.
The probabilities of innocent for 9, 8, 7 are:
0.2665677 = 0.00026 + ... + 0.16722 0.4059136 =0.00066 + ... + 0.23224 0.289792 = 0.00164 + ... + 0.19354
So, should go for 8 judges.
0.6 0.6 0.6 9 8 7 0 0.00026 0.00066 0.00164 1 0.00354 0.00786 0.01720 2 0.02123 0.04129 0.07741 3 0.07432 0.12386 0.19354 4 0.16722 0.23224 0.29030 5 0.25082 0.27869 0.26127 6 0.25082 0.20902 0.13064 7 0.16124 0.08958 0.02799 8 0.06047 0.01680 0.00000 9 0.01008 0.00000 0.00000
8.8(a) Prob(1) = 4*(.2)*(.8)^3 =0.4096
(b) E(X) = 0.4096*75 =30.72
Std Dev = sqrt(n*p*(1-p)) =4.258766
(c) Probabilities and expected #s are given.
It suggests that either nobody gets sick or 3 or 4 get sick.
Maybe some rooms don't get exposed, while most of the
patients who get exposed get sick. (p=0.83?)
0 0.40960 30.72 3 1 0.40960 30.72 2 2 0.15360 11.52 5 3 0.02560 1.92 17 4 0.00160 0.12 20
8.13 (a)
x p(x) 0 0.8836116 1 0.1128015 2 0.0035622 3 2.474E-05
(b) Skip
(c) Reject for 2 or 3. Prob = .003562+2.5e-5 =0.0035869
8.14 n=5 is smallest where Prob(0 males)+Prob(0 fem) < 0.10
n No males No fem. SUM Prob C(3,n) C(7,n) /C(10,n) 2 3 21 24 0.5333333 3 1 35 36 0.3 4 0 35 35 0.1666667 5 0 21 21 0.0833333 6 0 7 7 0.0333333 7 0 1 1 0.0083333
8.16 (a) Exact hypergeometric probabilities
N 40 15 8 x Hyper Binomial 0 0.0140638 0.0232831 1 0.0937589 0.1117587 2 0.2417993 0.2346933 3 0.3143391 0.2816319 4 0.2245279 0.211224 5 0.0898112 0.1013875 6 0.0195242 0.0304163 7 0.0020919 0.0052142 8 8.368E-05 0.0003911 SUM(5-8) 0.1115109 0.137409
(b)
100 15 8 x Hyper Binomial 0 0.2586117 0.2724905 1 0.3978642 0.3846925 2 0.2467765 0.2376042 3 0.0802024 0.0838603 4 0.0148523 0.0184986 5 0.0015939 0.0026116 6 9.602E-05 0.0002304 7 2.939E-06 1.162E-05 8 3.458E-08 2.563E-07 SUM(5-8) 0.0016929 0.0028539
(c)
150 15 8 0 0.4213107 0.4304672 1 0.3949788 0.3826375 2 0.1500307 0.1488035 3 0.0300061 0.0330674 4 0.0034358 0.0045927 5 0.0002291 0.0004082 6 8.611E-06 2.268E-05 7 1.652E-07 7.2E-07 8 1.224E-09 1E-08 SUM(5-8) 0.0002378 0.0004317
8.17 (a)
Hyper Bin. 300 135 12 6 0.2163025 0.2123847
(b)
7 0.1494805 0.1489451 8 0.074276 0.0761651 9 0.0258794 0.0276964 10 0.0060015 0.0067982 11 0.0008317 0.0010113 12 5.209E-05 6.895E-05 Pr(7-12) 0.2565212 0.2606851
8.18 (a) 2=lambda
x p(x) F(x) 0 0.1353353 0.1353353 1 0.2706706 0.4060058 2 0.2706706 0.6766764 3 0.180447 0.8571235 4 0.0902235 0.947347
Prob(X>=4) = 1-F(3) =0.1428765
(b) $200
(c) $100 or $200
(d) Avg cost @ $25/car = 25*2=$50
However, about 1/3 of the time, it will cost more @ $25/car
8.21 (a)
2=lambda
X p(x) Y p(y) E(Y) = 1.782 0 0.1353353 0 0.1353353 0 1 0.2706706 1 0.2706706 0.2706706 2 0.2706706 2 0.2706706 0.5413412 3 0.180447 3 0.3233236 0.9699707 4 0.0902235 3 1.7819824 5 0.0360894 3
8.26 Geometric, p=0.05
(a) E(X) = 1/p = 20
(b) P(X<=10) = F(10) = 1-(1-p)^10 =0.4012631
(c) F(x) = 1/2; 0.95^x=0.5
x=ln(0.5)/ln(0.95) =13.513407
x p(x) F(x) 1 0.05 0.05 2 0.0475 0.0975 3 0.045125 0.142625 4 0.0428688 0.1854938 5 0.0407253 0.2262191 6 0.038689 0.2649081 7 0.0367546 0.3016627 8 0.0349169 0.3365796 9 0.033171 0.3697506 10 0.0315125 0.4012631 11 0.0299368 0.4311999 12 0.02844 0.4596399 13 0.027018 0.4866579 14 0.0256671 0.512325 15 0.0243837 0.5367088 16 0.0231646 0.5598733 17 0.0220063 0.5818797
8.29 (a) Neg bin, p=0.5, r=
x p(x) F(x) 2 0.25 0.25 3 0.25 0.5 4 0.1875 0.6875 5 0.125 0.8125 6 0.078125 0.890625 7 0.046875 0.9375 8 0.0273438 0.9648437 9 0.015625 0.9804687
P(X=6)= 0.078125
P(X>3)= 0.25
P(X<=6)= 0.890625
(b) E(X)=2/p=4
(c) Most likely is 4
(d) E(boys)=E(children)-2 girls = 4-2 = 2
8.31 Neg bin, p=0.10, r=2
x p(x) F(x) 2 0.01 0.01 3 0.018 0.028 4 0.0243 0.0523 5 0.02916 0.08146 6 0.032805 0.114265 7 0.0354294 0.1496944 8 0.0372009 0.1868953 9 0.0382638 0.225159 10 0.038742 0.2639011 11 0.038742 0.3026431 12 0.0383546 0.3409977 13 0.0376573 0.378655 14 0.0367158 0.4153709 15 0.0355861 0.450957 16 0.0343152 0.4852722 17 0.0329426 0.5182148 18 0.0315013 0.5497161 19 0.0300189 0.579735 20 0.028518 0.608253 21 0.027017 0.63527
P(X=6)=0.032805
P(X<=6)=0.114265
P(X>4)=0.9477
(b) E(X)=2/p=20
E(bad)=20-2=18
(c) Median=16.5
(d) P(X>20)=0.391747
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