A manufacturer knows that on average 20% of toasters produced require repairs within 1 year after they are sold. When 18 toasters are randomly​ selected, find the smallest number x and largest number y such that​ (a) the probability that at least x of them will require repairs is less than 0.58; ​(b) the probability that at least y of them will not require repairs is greater than 0.84.

A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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A manufacturer knows that on average 20% of toasters produced require repairs within 1 year after they are sold. When 18 toasters are randomly​ selected, find the smallest number x and largest number y such that​ (a) the probability that at least x of them will require repairs is less than 0.58; ​(b) the probability that at least y of them will not require repairs is greater than 0.84.

Table A.1 (continued) Binomial Probability Sums b(a; n, p)
9
10
11
12
n
13
14
n r 0.10
0.20
19 0
0.1351 0.0144
0.4203 0.0829
1
0.1113
0.0462 0.0055 0.0004 0.0000
2 0.7054 0.2369
3 0.8850 0.4551
0.9648 0.6733
0.2631 0.1332
0.0230 0.0022 0.0001
4
0.0000
7
0.9997
8
1.0000
0.4654 0.2822 0.0696 0.0096 0.0006
5 0.9914 0.8369 0.6678 0.4739 0.1629 0.0318 0.0031 0.0001
6 0.9983 0.9324 0.8251 0.6655 0.3081 0.0835 0.0116 0.0006
0.9767 0.9225 0.8180 0.4878 0.1796
0.9933 0.9713 0.9161 0.6675 0.3238
0.9984 0.9911 0.9674 0.8139
0.9997 0.9977 0.9895 0.9115
1.0000 0.9995 0.9972 0.9648 0.8204
0.9999 0.9994 0.9884
0.0017
1.0000 0.9999 0.9969 0.9682 0.8371 0.5261 0.1631 0.0086
1.0000 0.9994 0.9904 0.9304 0.7178 0.3267 0.0352
0.9999 0.9978 0.9770 0.8668 0.5449 0.1150
1.0000 0.9996 0.9945 0.9538 0.7631 0.2946
1.0000 0.9992 0.9896 0.9171 0.5797
0.9999 0.9989 0.9856 0.8649
1.0000 1.0000 1.0000 1.0000
0.0352 0.0028 0.0000
0.0885 0.0105 0.0003
0.5000 0.1861 0.0326 0.0016
0.6762 0.3325 0.0839 0.0067 0.0000
0.5122 0.1820 0.0233 0.0003
0.9165 0.6919 0.3345 0.0676
15
16
17
18
19
20 0
10
11
12
13
14
15
16
17
18
19
20
r
0.10
r=0
0.25 0.30
0.40
0.0042
0.0011
0.0001
0.0310
0.0104 0.0008 0.0000
0.1216 0.0115
0.0032
0.0008 0.0000
1 0.3917 0.0692 0.0243 0.0076 0.0005
0.0000
2
0.0002
0.6769 0.2061 0.0913 0.0355 0.0036
0.8670 0.4114 0.2252 0.1071
0.0160
0.0013
0.0000
4
0.0510
0.0059
0.0003
0.9568 0.6296 0.4148 0.2375
0.9887 0.8042 0.6172
5
0.1256 0.0207 0.0016
0.0000
6
7 0.9996
8 0.9999
0.0000
9
1.0000
0.4164
0.9976 0.9133 0.7858 0.6080 0.2500 0.0577 0.0065 0.0003
0.9679 0.8982 0.7723 0.4159 0.1316 0.0210 0.0013
0.9900 0.9591 0.8867 0.5956 0.2517 0.0565 0.0051 0.0001
0.9974 0.9861 0.9520 0.7553 0.4119 0.1275 0.0171 0.0006
0.9994 0.9961 0.9829 0.8725 0.5881 0.2447 0.0480 0.0026 0.0000
0.9999 0.9991 0.9949 0.9435 0.7483 0.4044 0.1133 0.0100 0.0001
1.0000 0.9998 0.9987 0.9790 0.8684 0.5841 0.2277 0.0321 0.0004
1.0000 0.9997 0.9935 0.9423 0.7500 0.3920 0.0867 0.0024
0.9984 0.9793 0.8744 0.5836 0.1958
0.9997 0.9941 0.9490 0.7625 0.3704
1.0000
1.0000
0.0113
0.0432
0.9987 0.9840 0.8929 0.5886
0.1330
0.3231
0.9998
1.0000
0.6083
0.8784
1.0000
0.90
0.20 0.25
P
0.30
0.40
0.50
P
0.60
0.50
0.70
0.80
0.9964 0.9645 0.7939
0.9995 0.9924 0.9308
0.60
1.0000 0.9992 0.9885
1.0000
1.0000
0.70
0.80
0.90
Transcribed Image Text:Table A.1 (continued) Binomial Probability Sums b(a; n, p) 9 10 11 12 n 13 14 n r 0.10 0.20 19 0 0.1351 0.0144 0.4203 0.0829 1 0.1113 0.0462 0.0055 0.0004 0.0000 2 0.7054 0.2369 3 0.8850 0.4551 0.9648 0.6733 0.2631 0.1332 0.0230 0.0022 0.0001 4 0.0000 7 0.9997 8 1.0000 0.4654 0.2822 0.0696 0.0096 0.0006 5 0.9914 0.8369 0.6678 0.4739 0.1629 0.0318 0.0031 0.0001 6 0.9983 0.9324 0.8251 0.6655 0.3081 0.0835 0.0116 0.0006 0.9767 0.9225 0.8180 0.4878 0.1796 0.9933 0.9713 0.9161 0.6675 0.3238 0.9984 0.9911 0.9674 0.8139 0.9997 0.9977 0.9895 0.9115 1.0000 0.9995 0.9972 0.9648 0.8204 0.9999 0.9994 0.9884 0.0017 1.0000 0.9999 0.9969 0.9682 0.8371 0.5261 0.1631 0.0086 1.0000 0.9994 0.9904 0.9304 0.7178 0.3267 0.0352 0.9999 0.9978 0.9770 0.8668 0.5449 0.1150 1.0000 0.9996 0.9945 0.9538 0.7631 0.2946 1.0000 0.9992 0.9896 0.9171 0.5797 0.9999 0.9989 0.9856 0.8649 1.0000 1.0000 1.0000 1.0000 0.0352 0.0028 0.0000 0.0885 0.0105 0.0003 0.5000 0.1861 0.0326 0.0016 0.6762 0.3325 0.0839 0.0067 0.0000 0.5122 0.1820 0.0233 0.0003 0.9165 0.6919 0.3345 0.0676 15 16 17 18 19 20 0 10 11 12 13 14 15 16 17 18 19 20 r 0.10 r=0 0.25 0.30 0.40 0.0042 0.0011 0.0001 0.0310 0.0104 0.0008 0.0000 0.1216 0.0115 0.0032 0.0008 0.0000 1 0.3917 0.0692 0.0243 0.0076 0.0005 0.0000 2 0.0002 0.6769 0.2061 0.0913 0.0355 0.0036 0.8670 0.4114 0.2252 0.1071 0.0160 0.0013 0.0000 4 0.0510 0.0059 0.0003 0.9568 0.6296 0.4148 0.2375 0.9887 0.8042 0.6172 5 0.1256 0.0207 0.0016 0.0000 6 7 0.9996 8 0.9999 0.0000 9 1.0000 0.4164 0.9976 0.9133 0.7858 0.6080 0.2500 0.0577 0.0065 0.0003 0.9679 0.8982 0.7723 0.4159 0.1316 0.0210 0.0013 0.9900 0.9591 0.8867 0.5956 0.2517 0.0565 0.0051 0.0001 0.9974 0.9861 0.9520 0.7553 0.4119 0.1275 0.0171 0.0006 0.9994 0.9961 0.9829 0.8725 0.5881 0.2447 0.0480 0.0026 0.0000 0.9999 0.9991 0.9949 0.9435 0.7483 0.4044 0.1133 0.0100 0.0001 1.0000 0.9998 0.9987 0.9790 0.8684 0.5841 0.2277 0.0321 0.0004 1.0000 0.9997 0.9935 0.9423 0.7500 0.3920 0.0867 0.0024 0.9984 0.9793 0.8744 0.5836 0.1958 0.9997 0.9941 0.9490 0.7625 0.3704 1.0000 1.0000 0.0113 0.0432 0.9987 0.9840 0.8929 0.5886 0.1330 0.3231 0.9998 1.0000 0.6083 0.8784 1.0000 0.90 0.20 0.25 P 0.30 0.40 0.50 P 0.60 0.50 0.70 0.80 0.9964 0.9645 0.7939 0.9995 0.9924 0.9308 0.60 1.0000 0.9992 0.9885 1.0000 1.0000 0.70 0.80 0.90
Table A.1 (continued) Binomial Probability Sums b(x; n, p)
n r 0.10
17 0
n
7
8
9
10
11
0.20
0.1668 0.0225
0.0075
1 0.4818 0.1182 0.0501
2 0.7618 0.3096 0.1637 0.0774 0.0123 0.0012 0.0001
3 0.9174 0.5489 0.3530 0.2019 0.0464 0.0064 0.0005 0.0000
0.1260 0.0245 0.0025 0.0001
4
0.9779 0.7582 0.5739 0.3887
5 0.9953 0.8943 0.7653 0.5968
6 0.9992 0.9623 0.8929
0.9999 0.9891 0.9598
1.0000
12
13
14
15
16
17
18 0 0.1501 0.0180
10
11
12
13
14
15
16
17
18
r
0.10
0.25 0.30 0.40 0.50
0.0023 0.0002 0.0000
0.9974 0.9876 0.9597
0.9995 0.9969 0.9873
0.9999 0.9994
0.9999
1.0000
1.0000
0.20
x=0
0.25
0.0056
0.4503 0.0991 0.0395
1
2 0.7338 0.2713 0.1353 0.0600
3 0.9018 0.5010 0.3057 0.1646
4
5
0.0000
0.5634
0.9718 0.7164 0.5187 0.3327 0.0942 0.0154 0.0013 0.0000
0.9936 0.8671 0.7175 0.5344 0.2088 0.0481 0.0058 0.0003
6 0.9988 0.9487 0.8610 0.7217 0.3743 0.1189 0.0203 0.0014
7 0.9998 0.9837 0.9431 0.8593
0.0576 0.0061 0.0002
0.9957 0.9807 0.9404
0.9991 0.9946 0.9790
0.9998 0.9988 0.9939
1.0000 0.9998 0.9986
1.0000
0.2403
0.4073 0.1347 0.0210 0.0009
8
1.0000
0.7368
9
0.8653 0.5927 0.2632 0.0596 0.0043 0.0000
0.9424 0.7597 0.4366 0.1407 0.0163 0.0002
0.9797 0.8811 0.6257 0.2783 0.0513 0.0012
0.9942 0.9519 0.7912 0.4656 0.1329 0.0064
0.9987 0.9846 0.9058 0.6673 0.2836 0.0282
0.9998 0.9962 0.9672 0.8354 0.4990 0.0982
1.0000 0.9993 0.9918 0.9400 0.7287 0.2662
0.9999 0.9987 0.9858 0.9009 0.5497
1.0000 0.9999 0.9984 0.9820 0.8499
1.0000 1.0000 1.0000
1.0000
0.60 0.70 0.80 0.90
0.0193 0.0021 0.0001 0.0000
P
0.0016 0.0001
0.0142 0.0013
0.9997
1.0000
0.30
0.2639 0.0717 0.0106 0.0007 0.0000
0.7752 0.4478 0.1662 0.0348 0.0032 0.0001
0.8954 0.6405 0.3145 0.0919 0.0127 0.0005
0.8011 0.5000 0.1989 0.0403 0.0026 0.0000
0.9081 0.6855 0.3595 0.1046 0.0109 0.0001
0.9968 0.9652 0.8338 0.5522 0.2248 0.0377 0.0008
0.9993 0.9894 0.9283 0.7361 0.4032 0.1057 0.0047
0.9999 0.9975 0.9755 0.8740 0.6113 0.2418 0.0221
1.0000 0.9995 0.9936 0.9536 0.7981 0.4511 0.0826
0.9999 0.9988 0.9877 0.9226 0.6904 0.2382
1.0000 0.9999 0.9979 0.9807 0.8818 0.5182
1.0000 0.9998 0.9977 0.9775 0.8332
1.0000 1.0000 1.0000 1.0000
0.60
0.0000
0.0001
0.0082 0.0007 0.0000
0.0328 0.0038 0.0002
0.40
0.70
P
0.50
0.80
0.90
Transcribed Image Text:Table A.1 (continued) Binomial Probability Sums b(x; n, p) n r 0.10 17 0 n 7 8 9 10 11 0.20 0.1668 0.0225 0.0075 1 0.4818 0.1182 0.0501 2 0.7618 0.3096 0.1637 0.0774 0.0123 0.0012 0.0001 3 0.9174 0.5489 0.3530 0.2019 0.0464 0.0064 0.0005 0.0000 0.1260 0.0245 0.0025 0.0001 4 0.9779 0.7582 0.5739 0.3887 5 0.9953 0.8943 0.7653 0.5968 6 0.9992 0.9623 0.8929 0.9999 0.9891 0.9598 1.0000 12 13 14 15 16 17 18 0 0.1501 0.0180 10 11 12 13 14 15 16 17 18 r 0.10 0.25 0.30 0.40 0.50 0.0023 0.0002 0.0000 0.9974 0.9876 0.9597 0.9995 0.9969 0.9873 0.9999 0.9994 0.9999 1.0000 1.0000 0.20 x=0 0.25 0.0056 0.4503 0.0991 0.0395 1 2 0.7338 0.2713 0.1353 0.0600 3 0.9018 0.5010 0.3057 0.1646 4 5 0.0000 0.5634 0.9718 0.7164 0.5187 0.3327 0.0942 0.0154 0.0013 0.0000 0.9936 0.8671 0.7175 0.5344 0.2088 0.0481 0.0058 0.0003 6 0.9988 0.9487 0.8610 0.7217 0.3743 0.1189 0.0203 0.0014 7 0.9998 0.9837 0.9431 0.8593 0.0576 0.0061 0.0002 0.9957 0.9807 0.9404 0.9991 0.9946 0.9790 0.9998 0.9988 0.9939 1.0000 0.9998 0.9986 1.0000 0.2403 0.4073 0.1347 0.0210 0.0009 8 1.0000 0.7368 9 0.8653 0.5927 0.2632 0.0596 0.0043 0.0000 0.9424 0.7597 0.4366 0.1407 0.0163 0.0002 0.9797 0.8811 0.6257 0.2783 0.0513 0.0012 0.9942 0.9519 0.7912 0.4656 0.1329 0.0064 0.9987 0.9846 0.9058 0.6673 0.2836 0.0282 0.9998 0.9962 0.9672 0.8354 0.4990 0.0982 1.0000 0.9993 0.9918 0.9400 0.7287 0.2662 0.9999 0.9987 0.9858 0.9009 0.5497 1.0000 0.9999 0.9984 0.9820 0.8499 1.0000 1.0000 1.0000 1.0000 0.60 0.70 0.80 0.90 0.0193 0.0021 0.0001 0.0000 P 0.0016 0.0001 0.0142 0.0013 0.9997 1.0000 0.30 0.2639 0.0717 0.0106 0.0007 0.0000 0.7752 0.4478 0.1662 0.0348 0.0032 0.0001 0.8954 0.6405 0.3145 0.0919 0.0127 0.0005 0.8011 0.5000 0.1989 0.0403 0.0026 0.0000 0.9081 0.6855 0.3595 0.1046 0.0109 0.0001 0.9968 0.9652 0.8338 0.5522 0.2248 0.0377 0.0008 0.9993 0.9894 0.9283 0.7361 0.4032 0.1057 0.0047 0.9999 0.9975 0.9755 0.8740 0.6113 0.2418 0.0221 1.0000 0.9995 0.9936 0.9536 0.7981 0.4511 0.0826 0.9999 0.9988 0.9877 0.9226 0.6904 0.2382 1.0000 0.9999 0.9979 0.9807 0.8818 0.5182 1.0000 0.9998 0.9977 0.9775 0.8332 1.0000 1.0000 1.0000 1.0000 0.60 0.0000 0.0001 0.0082 0.0007 0.0000 0.0328 0.0038 0.0002 0.40 0.70 P 0.50 0.80 0.90
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