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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.

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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 9E
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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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