MSA Electronics is experimenting with the manufacture of a new type of transistor that is very difficult to mass-produce at an acceptable quality level. Every hour a supervisor takes a random sample of 6 transistors produced on the assembly line. The probability that any one transistor is defective is considered to be .13. MSA wants to know the proba-bility of finding 4 or more defects in the lot sampled.
The elements in this problem would be:
p .13 r 4 defects n 6 trials
The question posed may be easily answered by using a cumulative binomial distribu-tion table. Such tables can be very lengthy. For the sake of brevity, we present in Table M6.2 only that portion of a binomial table corresponding to n 6. Other books may con-tain complete binomial tables for a broad range of n, r, and p values.
Since the probability of MSA finding any one defect is .13, we look through the n 6 table until we find the column where p .13. We then move down that column until we are opposite the r 4 row. The answer there is found to be .0034, which is a probability of .0034 that there are 4 or more defects in the sample. This value has been shaded in Table M6.2.
M6-4 Module 6 TH E BI N O M I A L DI S T R I B U T I O N
P(R rn, p)
P .01 .02 .03 .04 .05 .06 .07 .08 .09 .10
R
1 .0585 .1142 .1670 .2172 .2649 .3101 .3530 .3936 .4321 .4686 2 .0015 .0057 .0125 .0216 .0328 .0459 .0608 .0773 .0952 .1143
3 .0002 .0005 .0012 .0022 .0038 .0058 .0085 .0118 .0159
4 .0001 .0002 .0003 .0005 .0008 .0013
5 .0001
P .11 .12 .13 .14 .15 .16 .17 .18 .19 .20
R
1 .5030 .5356 .5664 .5954 .6229 .6487 .6731 .6960 .7176 .7397 2 .1345 .1556 .1776 .2003 .2235 .2472 .2713 .2956 .3201 .3446 3 .0206 .0261 .0324 .0395 .0473 .0560 .0655 .0759 .0870 .0989 4 .0018 .0025 .0034 .0045 .0059 .0075 .0094 .0116 .0141 .0170 5 .0001 .0001 .0002 .0003 .0004 .0005 .0007 .0010 .0013 .0016
6 .0001
T A B L E M 6 . 2 A Sample Table for the Cumulative Binomial Distribution for n=6
(continued)
Expected Value and Variance There is an easy way to compute the expected value and variance of the binomial distribution. The appropriate equations are:
(M6-2) (M6-3)
The expected value and variance for MSA Electronics can be computed as follows:
(6)(.13)(1 .13) .6786 Variance np(1 p)
(6)(.13) .78 Expected value np
Variance np(1 p) Expected value np
M6.3 Solving Problems with Binomial Tables M6-5
P .21 .22 .23 .24 .25 .26 .27 .28 .29 .30
R
1 .7569 .7748 .7916 .8073 .8220 .8358 .8487 .8607 .8719 .8824 2 .3692 .3937 .4180 .4422 .4661 .4896 .5128 .5356 .5580 .5798 3 .1115 .1250 .1391 .1539 .1694 .1856 .2023 .2196 .2374 .2557 4 .0202 .0239 .0280 .0326 .0376 .0431 .0492 .0557 .0628 .0705 5 .0020 .0025 .0031 .0038 .0046 .0056 .0067 .0079 .0093 .0109 6 .0001 .0001 .0001 .0002 .0002 .0003 .0004 .0005 .0006 .0007
P .31 .32 .33 .34 .35 .36 .37 .38 .39 .40
R
1 .8921 .9011 .9095 .9173 .9246 .9313 .9375 .9432 .9485 .9533 2 .6012 .6220 .6422 .6619 .6809 .6994 .7172 .7343 .7508 .7667 3 .2744 .2936 .3130 .3328 .3529 .3732 .3937 .4143 .4350 .4557 4 .0787 .0875 .0969 .1069 .1174 .1286 .1404 .1527 .1657 .1792 5 .0127 .0148 .0170 .0195 .0223 .0254 .0288 .0325 .0365 .0410 6 .0009 .0011 .0013 .0015 .0018 .0022 .0026 .0030 .0035 .0041
P .41 .42 .43 .44 .45 .46 .47 .48 .49 .50
R
1 .9578 .9619 .9657 .9692 .9723 .9752 .9778 .9802 .9824 .9844 2 .7819 .7965 .8105 .8238 .8364 .8485 .8599 .8707 .8810 .8906 3 .4764 .4971 .5177 .5382 .5585 .5786 .5985 .6180 .6373 .6563 4 .1933 .2080 .2232 .2390 .2553 .2721 .2893 .3070 .3252 .3438 5 .0458 .0510 .0566 .0627 .0692 .0762 .0837 .0917 .1003 .1094 6 .0048 .0055 .0063 .0073 .0083 .0095 .0108 .0122 .0138 .0156
Source: Reprinted from Robert O. Schlaifer, Introduction to Statistics for Business Decisions, published by McGraw-Hill Book Company, 1961, by permission of the copyright holder, the President and Fellows of Harvard College.
T A B L E M 6 . 2 (continued)
Discussion Questions and Problems Discussion Questions
M6-1 What is the Bernoulli process? What probability distribution describes the Bernoulli process, and what conditions must be satisfied before this distribution can be used?
M6-2 What type of distribution is the binomial distribution? What type of distribution is the normal distribution?
Problems
M6-3 This year, Jan Rich, who is ranked number one in women’s singles in tennis, and Marie Wacker, who is ranked number three, will play 4 times. If Marie can beat Jan 3 times, she will be ranked number one. The two players have played 20 times before, and Jan has won 15 games. It is expected that this pattern will continue in the future. What is the probability that Marie will be ranked number one after this year? What is the probability that Marie will win all 4 games this year against Jan?
M6-4 Over the last two months, the Wilmington Phantoms have been encountering trouble with one of their star basketball players. During the last 30 games, he has fouled out 15 times. The owner of the basketball team has stated that if this player fouls out 2 times in their next 5 games, the player will be fined $200. What is the probability that the player will be fined? What is the probability that the player will foul out of all 5 games? What is the probability that the player will not foul out of any of the next 5 games?
M6-5 Wisconsin Cheese Processor, Inc., produces equipment that processes cheese products.
Ken Newgren is particularly concerned about a new cheese processor that has been pro-ducing defective cheese crocks. The piece of equipment produces 5 cheese crocks dur-ing every cycle of the equipment. The probability that any one of the cheese crocks is defective is .2. Ken would like to determine the probability distribution of defective cheese crocks from this new piece of equipment. There can be 0, 1, 2, 3, 4, or 5 defec-tive cheese crocks for any cycle of the equipment.
M6-6 Refer to Problem M6-5. Determine the expected value and variance of the distribution described in Problem M6-3, using Equations M6-2 and M6-3.
M6-7 Natway, a national distribution company of home vacuum cleaners, recommends that its salespersons make only two calls per day, one in the morning and one in the afternoon.
Twenty-five percent of the time a sales call will result in a sale, and the profit from each sale is $125.
(a) Develop the probability distribution for sales during a five-day week.
(b) Determine the mean and variance of this distribution.
(c) What is the expected weekly profit for a salesperson?
M6-6 Module 6 TH E BI N O M I A L DI S T R I B U T I O N
Case Study WTVX
WTVX, Channel 6, is located in Eugene, Oregon, home of the University of Oregon’s football team. The station was owned and operated by George Wilcox, a former Duck (University of Oregon football player). Although there were other television stations in Eugene, WTVX was the only station that had a weath-erperson who was a member of the American Meteorological Society (AMS). Every night, Joe Hummel would be introduced as the only weatherperson in Eugene who was a member of the AMS. This was George’s idea, and he believed that this gave his station the mark of quality and helped with market share.
In addition to being a member of AMS, Joe was also the most popular person on any of the local news programs. Joe was always trying to find innovative ways to make the weather interesting, and this was especially difficult during the winter months when the weather seemed to remain the same over long periods of time. Joe’s forecast for next month, for exam-ple, was that there would be a 70% chance of rain every day, and that what happens on one day (rain or shine) was not in any way dependent on what happened the day before.
One of Joe’s most popular features of the weather report was to invite questions during the actual broadcast.
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Case Study M6-7
tions would be phoned in, and they were answered on the spot by Joe. Once a ten-year-old boy asked what caused fog, and Joe did an excellent job of describing some of the vari-ous causes.
Occasionally, Joe would make a mistake. For example, a high school senior asked Joe what the chances were of getting 15 days of rain in the next month (30 days). Joe made a quick calculation: (70%) (15 days/30 days) (70%)(Z\x) 35%.
Joe quickly found out what it was like being wrong in a uni-versity town. He had over 50 phone calls from scientists, mathematicians, and other university professors, telling him
that he had made a big mistake in computing the chances of getting 15 days of rain during the next 30 days. Although Joe didn’t understand all of the formulas the professors mentioned, he was determined to find the correct answer and make a cor-rection during a future broadcast.
Discussion Questions
1. What are the chances of getting 15 days of rain during the next 30 days?
2. What do you think about Joe’s assumptions concerning the weather for the next 30 days?