Bayes’ Theorem
Case 4.2 Pittsburg Lighting
Melinda Wilson and Tony Green left their boss’s office, heading for the conference room to discuss the task they were given. The task is to select one of two suppliers of outdoor light bulbs. Ordi- narily this would be pretty easy—just pick the one that offers the best price. But in this case, their boss is planning to provide a war- ranty on the bulbs that will require the company to replace all defects at no cost to the customer. Thus, product quality is an important consideration.
As Melinda and Tony settled into the plush leather conference room seats, they quickly realized that this task was more
challenging than they first thought. To get started, Melinda went to the white board and began listing the information they already knew. The total purchase will be 100,000 bulbs. Regardless of which supplier is selected, the bulbs will be sold to Pittsburg cus- tomers at the same price. The first supplier, Altman Electronics, was one that Pittsburg had used for years, so they had solid quality information. Both Melinda and Tony agreed that this supplier could be counted on for a 5% defect rate. Altman’s bid was $3.00 per light. The second supplier, Crestwell Lights and Fixtures, was new to Pittsburg but offered an attractive bid of $2.70 per unit.
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Case 4.2 Pittsburg Lighting
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Chapter 4 195Possible Defect
Rate Probability of Six Defects Given the Possible Defect Rate
0.02 0.01
0.03 0.05
0.04 0.11
0.05 0.15
0.06 0.17
0.07 0.15
0.08 0.12
Melinda and Tony returned to the conference room to consider all this information and make a decision regarding which supplier to select. They agreed that they would recommend the supplier that would provide the lowest overall cost, including purchasing and warranty replacement costs.
Required Tasks:
1. Calculate the total purchase cost for each supplier.
2. Calculate the warranty replacement cost for Altman Electronics, and calculate the total cost if Altman is selected.
3. Calculate the total cost (purchase cost plus defective replacement) for each possible defect level if Crestwell is selected.
4. Based on the findings from the order of 100 bulbs from Crestwell, use Bayes’ Theorem to revise Tony’s subjective probability assessments for Crestwell’s possible defect rates.
5. Using the revised probabilities, calculate the weighted total costs for Crestwell, and compare this to the total costs calculated for Altman. Which supplier should Melinda and Tony recommend?
However, neither Melinda nor Tony had any information about Crestwell’s likely defect rate. They realized that they needed to get some more information before they could proceed.
After lunch, Tony made a few calls to other lighting companies in different parts of the country to see what their experience was with Crestwell. When Tony met Melinda back in the conference room, he went to the board and wrote his findings:
Crestwell Defect Rate Probability
0.02 0.05
0.03 0.10
0.04 0.20
0.05 0.25
0.06 0.20
0.07 0.15
0.08 0.05
Tony explained to Melinda that Crestwell’s defect performance differed depending on which person he talked to, but these numbers summarized his findings. Thus, Crestwell’s defect rate is uncertain, and the probability of any specific defect level is really a subjective assessment made by Tony. Because customers would be refunded for defective bulbs, both Melinda and Tony were concerned that they could not determine a particular defect rate. Melinda sug- gested that they place a small order for 100 bulbs and see what the defect rate for those turned out to be. She figured that they could use this added information to learn more about Crestwell’s quality.
A few days later, Pittsburg received the 100 bulbs from Crestwell, and after the warehouse crew tested them, six were found to be defec- tive. Tony did some quick calculations and found the following:
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196
WHY YOU NEED TO KNOW
How many toll stations should be constructed when a new toll bridge is built? If there are four toll stations, will drivers have to wait too long, or will there be too many toll stations and excess employees? To help answer these questions, decision makers use a probability distribution known as the Poisson distribution.
Pak-Sense, a manufacturer of temperature sensor equipment for the food industry, receives component parts for its sensors weekly from suppliers. When a batch of parts arrives, the quality-assurance section randomly samples a fixed number of parts and tests them to see
Discrete Probability Distributions
5
Introduction to Discrete Probability Distributions
(pg. 197–204)
o u t c o m e 1 Be able to calculate and interpret the expected value of a discrete random variable.
The Binomial Probability Distribution (pg. 204–216)
o u t c o m e 2 Be able to apply the binomial distribution to business decision-making situations.
Other Probability Distributions (pg. 217–228)
o u t c o m e 3 Be able to compute probabilities for the Poisson and hypergeometric distributions and apply these distributions to decision-making situations.
5.1
Review the concepts of simple random sampling discussed in Chapter 1.
Review the discussion of weighted averages in Chapter 3.
Review the basic rules of probability in Chapter 4, including the Addition and Multiplication Rules.
196
Quick Prep 5.2
5.3
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5.1 Introduction to Discrete Probability Distributions
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Chapter 5 197if any are defective. Suppose in one such test, a sample of 50 parts is selected from a supplier whose contract calls for at most 2% defective parts. How many defective parts in the sample of 50 should Pak-Sense expect if the contract is being satisfied? What should be concluded if the sample contains three defects? Answers to these questions require calculations based on a probability distribution known as the binomial distribution.
A personnel manager has a chance to promote three people from ten equally qualified candidates. Suppose none of six women are selected by the manager. Is this evidence of gender bias, or would we expect to see this type of result? A distribution known as the hypergeometric distribution would be very helpful in addressing this issue.
The binomial, Poisson, and hypergeometric distributions are three discrete probability distributions used in business decision making. This chapter introduces discrete probability distributions and shows how they are used in business settings. Through the use of these well- established discrete probability distributions, you will be better prepared for making decisions in an uncertain environment.
5.1 Introduction to Discrete Probability Distributions Random Variables
As discussed in Chapter 4, when a random experiment is performed, some outcome must occur. When the experiment has a quantitative characteristic, we can associate a number with each outcome. For example, an inspector who examines three HD televisions can judge each television as “acceptable” or “unacceptable.” The outcome of the experiment defines the spe- cific number of acceptable televisions. The possible outcomes are
x = 50, 1, 2, 36
The value x is called a random variable, since the numerical values it takes on are random and vary from trial to trial. Although the inspector knows these are the possible values for the variable before she samples, she does not know which value will occur in any given trial.
Further, the value of the random variable may be different each time three HD televisions are inspected.
Two classes of random variables exist: discrete random variables and continuous random variables. For instance, if a bank auditor randomly examines 15 accounts to verify the accuracy of the balances, the number of inaccurate account balances can be represented by a discrete random variable with the following values:
x = 50, 1, . . . ., 156
In another situation, ten employees were recently hired by a major electronics company. The number of females in that group can be described as a discrete random variable with possible values equal to
x = 50, 1, 2, 3, . . . ., 106
Notice that the value for a discrete random variable is often determined by counting. In the bank auditing example, the value of variable x is determined by counting the number of accounts with errors. In the hiring example, the value of variable x is determined by counting the number of females hired.
In other situations, the random variable is said to be continuous. For example, the exact time it takes a city bus to complete its route may be any value between two points, say 30 min- utes to 35 minutes. If x is the time required, then x is continuous because, if measured precisely enough, the possible values, x, can be any value in the interval 30 to 35 minutes. Other examples of continuous variables include measures of distance and measures of weight when measured precisely. A continuous random variable is generally defined by measuring, which is contrasted with a discrete random variable, whose value is typically determined by counting. Chapter 6 focuses on some important probability distributions for continuous random variables.
Random Variable
A variable that takes on different numerical values based on chance.
Discrete Random Variable A random variable that can assume only a finite number of values or an infinite sequence of values such as 0, 1, 2, . . . .
Continuous Random Variables Random variables that can assume an uncountably infinite number of values.
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198 Chapter 5
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Discrete Probability DistributionsDisplaying Discrete Probability Distributions Graphically The probability dis- tribution for a discrete random variable is composed of the values the variable can assume and the probabilities for each of the possible values. For example, if three parts are tested to determine whether they are defective, the probability distribution for the number of defectives might be
x = Number of Defectives P(x)
0 0.10
1 0.30
2 0.40
3 0.20
a = 1.00
Graphically, the discrete probability distribution associated with these defectives can be represented by the areas of rectangles in which the base of each rectangle is one unit wide and the height corresponds to the probability. The areas of the rectangles sum to 1. Figure 5.1 illustrates two examples of discrete probability distributions. Figure 5.1(a) shows a discrete random variable with only three possible outcomes. Figure 5.1(b) shows the probability dis- tribution for a discrete variable that has 21 possible outcomes. Note that as the number of possible outcomes increases, the distribution becomes smoother and the individual probabil- ity of any particular value tends to be reduced. In all cases, the sum of the probabilities is 1.
Discrete probability distributions have many applications in business decision-making situations. In the remainder of this section, we discuss several issues that are of particular importance to discrete probability distributions.
Probability
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Possible Values of x
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0
(b) Discrete Probability Distribution (21 possible outcomes)
x P(x)
Probability
10 20 30
Possible Values of x 0.1
0.2 0.3 0.4 0.5 0.6 0.7
0
(a) Discrete Probability Distribution (3 possible outcomes)
x P(x)
FIGURE 5.1 Discrete Probability Distributions
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5.1 Introduction to Discrete Probability Distributions