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(x;n, p)n T 0.100.2017 0 0.1668 0.02251 0.4818 0.11822 0.7618 0.3096 0.16373 0.9174 0.5489 0.3530 0.20190.9779 0.7582 0.5739 0.3887450.9953 0.8943 0.765360.999281.00000.00470.9623 0.8929 0.7752 0.4478 0.1662 0.0348 0.0032 0.00017 0.9999 0.9891 0.9598 0.8954 0.6405 0.3145 0.0919 0.0127 0.00050.9974 0.9876 0.9597 0.8011 0.5000 0.1989 0.0403 0.0026 0.00000.9995 0.9969 0.9873 0.9081 0.6855 0.3595 0.1046 0.0109 0.00010.9999 0.9994 0.9968 0.9652 0.8338 0.5522 0.2248 0.0377 0.00081.0000 0.9999 0.9993 0.9894 0.9283 0.7361 0.4032 0.10571.0000 0.9999 0.9975 0.9755 0.8740 0.6113 0.2418 0.02211.0000 0.9995 0.9936 0.9536 0.7981 0.4511 0.08260.9999 0.9988 0.9877 0.9226 0.6904 0.23821.0000 0.9999 0.9979 0.9807 0.8818 0.51820.9998 0.9977 0.9775 0.83321.0000 1.0000 1.0000 1.00001.00009101112n131415161718 0 0.15011231011121314151617180.0180 0.00560.00160.00010.00000.4503 0.0991 0.03950.0142 0.00130.00010.7338 0.2713 0.1353 0.0600 0.00820.00070.9018 0.5010 0.3057 0.1646 0.0328 0.003840.97185 0.9936 0.8671 0.717560.7164 0.5187 0.3327 0.0942 0.0154 0.0013 0.00000.5344 0.2088 0.0481 0.0058 0.00030.9988 0.9487 0.8610 0.7217 0.37430.9837 0.9431 0.8593 0.56340.9957 0.980770.999881.000090.99910.99981.0000P 0.100.400.25 0.300.0075 0.0023 0.00020.05010.0193 0.00210.20x=00.500.00000.0001 0.00000.0774 0.0123 0.0012 0.00010.250.300.600.0464 0.0064 0.0005 0.00000.1260 0.0245 0.0025 0.00010.5968 0.2639 0.0717 0.0106 0.0007 0.00000.400.70P0.1189 0.0203 0.0014 0.00000.2403 0.0576 0.0061 0.00020.4073 0.1347 0.0210 0.00090.9998 0.99861.0000 0.99971.00000.9404 0.73680.9946 0.9790 0.8653 0.5927 0.2632 0.0596 0.0043 0.00000.9988 0.9939 0.9424 0.7597 0.4366 0.1407 0.0163 0.00020.9797 0.8811 0.6257 0.2783 0.0513 0.00120.9942 0.9519 0.7912 0.4656 0.1329 0.00640.9987 0.9846 0.9058 0.6673 0.2836 0.02820.9998 0.9962 0.9672 0.8354 0.4990 0.09821.0000 0.9993 0.9918 0.9400 0.7287 0.26620.9858 0.90090.54970.9999 0.99871.0000 0.99990.9984 0.98200.84991.0000 1.0000 1.00001.00000.60 0.700.800.900.500.00000.00020.800.90 Table A.1 (continued) Binomial Probability Sums b(x; n, p)n0.200.25 0.30 0.4019 00.13510.01440.00420.00110.000110.42030.08290.03100.01040.00082 0.70540.23690.1113 0.0462 0.00553 0.88500.45510.2631 0.1332 0.02300.6733 0.4654 0.2822 0.06965 0.9914 0.8369 0.6678 0.4739 0.16294 0.9648P 0.10n789101112131415161718191011121314151617181920r0.0835 0.0116 0.00060.1796 0.0352 0.0028 0.00000.0885 0.0105 0.00030.5000 0.1861 0.03266 0.9983 0.9324 0.8251 0.6655 0.30810.9997 0.9767 0.9225 0.8180 0.48781.0000 0.9933 0.9713 0.9161 0.6675 0.32380.9984 0.9911 0.9674 0.81390.00160.9997 0.9977 0.9895 0.9115 0.6762 0.3325 0.0839 0.0067 0.00001.0000 0.9995 0.9972 0.9648 0.8204 0.5122 0.1820 0.0233 0.00030.9999 0.9994 0.9884 0.9165 0.6919 0.3345 0.0676 0.00171.0000 0.9999 0.9969 0.9682 0.8371 0.5261 0.1631 0.00861.0000 0.9994 0.9904 0.9304 0.7178 0.32670.9999 0.9978 0.9770 0.8668 0.54491.00000.03520.11500.99960.9945 0.9538 0.76310.29461.00000.9992 0.9896 0.9171 0.57970.9999 0.9989 0.9856 0.86491.0000 1.00001.00001.00000.01150.3917 0.06920.6769 0.2061 0.09132=00.00320.0243 0.007620 0 0.121610.0005 0.000020.0355 0.0036 0.00020.00003 0.8670 0.4114 0.2252 0.1071 0.0160 0.00134 0.9568 0.6296 0.4148 0.2375 0.0510 0.0059 0.00035 0.9887 0.8042 0.617260.41640.9976 0.9133 0.7858 0.60800.9996 0.9679 0.8982 0.77230.99999 1.0000780.100.0008 0.0000P0.20 0.25 0.300.500.400.60P0.00000.0004 0.00000.0022 0.00010.0096 0.0006 0.00000.0318 0.0031 0.00010.1256 0.0207 0.0016 0.00000.25000.0577 0.0065 0.00030.1316 0.0210 0.0013 0.00000.2517 0.0565 0.0051 0.00010.4119 0.1275 0.0171 0.00060.41590.9900 0.9591 0.8867 0.59560.9974 0.9861 0.9520 0.75530.9994 0.9961 0.9829 0.8725 0.5881 0.2447 0.0480 0.0026 0.00000.9999 0.9991 0.9949 0.9435 0.7483 0.4044 0.1133 0.0100 0.00011.0000 0.9998 0.9987 0.9790 0.8684 0.5841 0.2277 0.0321 0.00041.0000 0.9997 0.9935 0.9423 0.7500 0.3920 0.0867 0.00241.0000 0.9984 0.9793 0.8744 0.5836 0.1958 0.01130.9997 0.9941 0.9490 0.7625 0.3704 0.04321.0000 0.9987 0.9840 0.8929 0.5886 0.13300.9998 0.9964 0.9645 0.7939 0.32311.0000 0.9995 0.9924 0.9308 0.60831.0000 0.9992 0.9885 0.87841.0000 1.0000 1.00000.70 0.80 0.900.70 0.800.500.900.60

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.

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Table A.1 (continued) Binomial Probability Sums b(x;n, p)
n T 0.10
0.20
17 0 0.1668 0.0225
1 0.4818 0.1182
2 0.7618 0.3096 0.1637
3 0.9174 0.5489 0.3530 0.2019
0.9779 0.7582 0.5739 0.3887
4
5
0.9953 0.8943 0.7653
6
0.9992
8
1.0000
0.0047
0.9623 0.8929 0.7752 0.4478 0.1662 0.0348 0.0032 0.0001
7 0.9999 0.9891 0.9598 0.8954 0.6405 0.3145 0.0919 0.0127 0.0005
0.9974 0.9876 0.9597 0.8011 0.5000 0.1989 0.0403 0.0026 0.0000
0.9995 0.9969 0.9873 0.9081 0.6855 0.3595 0.1046 0.0109 0.0001
0.9999 0.9994 0.9968 0.9652 0.8338 0.5522 0.2248 0.0377 0.0008
1.0000 0.9999 0.9993 0.9894 0.9283 0.7361 0.4032 0.1057
1.0000 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
0.9998 0.9977 0.9775 0.8332
1.0000 1.0000 1.0000 1.0000
1.0000
9
10
11
12
n
13
14
15
16
17
18 0 0.1501
1
2
3
10
11
12
13
14
15
16
17
18
0.0180 0.0056
0.0016
0.0001
0.0000
0.4503 0.0991 0.0395
0.0142 0.0013
0.0001
0.7338 0.2713 0.1353 0.0600 0.0082
0.0007
0.9018 0.5010 0.3057 0.1646 0.0328 0.0038
4
0.9718
5 0.9936 0.8671 0.7175
6
0.7164 0.5187 0.3327 0.0942 0.0154 0.0013 0.0000
0.5344 0.2088 0.0481 0.0058 0.0003
0.9988 0.9487 0.8610 0.7217 0.3743
0.9837 0.9431 0.8593 0.5634
0.9957 0.9807
7
0.9998
8
1.0000
9
0.9991
0.9998
1.0000
P 0.10
0.40
0.25 0.30
0.0075 0.0023 0.0002
0.0501
0.0193 0.0021
0.20
x=0
0.50
0.0000
0.0001 0.0000
0.0774 0.0123 0.0012 0.0001
0.25
0.30
0.60
0.0464 0.0064 0.0005 0.0000
0.1260 0.0245 0.0025 0.0001
0.5968 0.2639 0.0717 0.0106 0.0007 0.0000
0.40
0.70
P
0.1189 0.0203 0.0014 0.0000
0.2403 0.0576 0.0061 0.0002
0.4073 0.1347 0.0210 0.0009
0.9998 0.9986
1.0000 0.9997
1.0000
0.9404 0.7368
0.9946 0.9790 0.8653 0.5927 0.2632 0.0596 0.0043 0.0000
0.9988 0.9939 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
0.9858 0.9009
0.5497
0.9999 0.9987
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
0.50
0.0000
0.0002
0.80
0.90

Table A.1 (continued) Binomial Probability Sums b(x; n, p)
n
0.20
0.25 0.30 0.40
19 0
0.1351
0.0144
0.0042
0.0011
0.0001
1
0.4203
0.0829
0.0310
0.0104
0.0008
2 0.7054
0.2369
0.1113 0.0462 0.0055
3 0.8850
0.4551
0.2631 0.1332 0.0230
0.6733 0.4654 0.2822 0.0696
5 0.9914 0.8369 0.6678 0.4739 0.1629
4 0.9648
P 0.10
n
7
8
9
10
11
12
13
14
15
16
17
18
19
10
11
12
13
14
15
16
17
18
19
20
r
0.0835 0.0116 0.0006
0.1796 0.0352 0.0028 0.0000
0.0885 0.0105 0.0003
0.5000 0.1861 0.0326
6 0.9983 0.9324 0.8251 0.6655 0.3081
0.9997 0.9767 0.9225 0.8180 0.4878
1.0000 0.9933 0.9713 0.9161 0.6675 0.3238
0.9984 0.9911 0.9674 0.8139
0.0016
0.9997 0.9977 0.9895 0.9115 0.6762 0.3325 0.0839 0.0067 0.0000
1.0000 0.9995 0.9972 0.9648 0.8204 0.5122 0.1820 0.0233 0.0003
0.9999 0.9994 0.9884 0.9165 0.6919 0.3345 0.0676 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
0.9999 0.9978 0.9770 0.8668 0.5449
1.0000
0.0352
0.1150
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
0.0115
0.3917 0.0692
0.6769 0.2061 0.0913
2=0
0.0032
0.0243 0.0076
20 0 0.1216
1
0.0005 0.0000
2
0.0355 0.0036 0.0002
0.0000
3 0.8670 0.4114 0.2252 0.1071 0.0160 0.0013
4 0.9568 0.6296 0.4148 0.2375 0.0510 0.0059 0.0003
5 0.9887 0.8042 0.6172
6
0.4164
0.9976 0.9133 0.7858 0.6080
0.9996 0.9679 0.8982 0.7723
0.9999
9 1.0000
7
8
0.10
0.0008 0.0000
P
0.20 0.25 0.30
0.50
0.40
0.60
P
0.0000
0.0004 0.0000
0.0022 0.0001
0.0096 0.0006 0.0000
0.0318 0.0031 0.0001
0.1256 0.0207 0.0016 0.0000
0.2500
0.0577 0.0065 0.0003
0.1316 0.0210 0.0013 0.0000
0.2517 0.0565 0.0051 0.0001
0.4119 0.1275 0.0171 0.0006
0.4159
0.9900 0.9591 0.8867 0.5956
0.9974 0.9861 0.9520 0.7553
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
1.0000 0.9984 0.9793 0.8744 0.5836 0.1958 0.0113
0.9997 0.9941 0.9490 0.7625 0.3704 0.0432
1.0000 0.9987 0.9840 0.8929 0.5886 0.1330
0.9998 0.9964 0.9645 0.7939 0.3231
1.0000 0.9995 0.9924 0.9308 0.6083
1.0000 0.9992 0.9885 0.8784
1.0000 1.0000 1.0000
0.70 0.80 0.90
0.70 0.80
0.50
0.90
0.60

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