Can businesses use probabilities to improve business decisions in the organization? Please explain with examples. 2. Let’s say you are part of a manufacturing organization, and you w
Use the information in the book or other sources to complete the following questions:
1. Can businesses use probabilities to improve business decisions in the organization? Please explain with examples.
2. Let's say you are part of a manufacturing organization, and you want to open another line of business. You received news of three possibilities. The line will have a 45% chance of making a profit of $150,000, 20% chance of making $25,000, and a 30% chance of losing $125,000. What is your decision? Think about objective and subjective probabilities.
3. For this question, you will need to understand Poisson's distribution. You own an ice cream shop selling 4 ice cream sandwiches every hour. You believe that you can sell 6 ice cream sandwiches in the next hour. What is the probability of this happening? Explain and show your work.
Need about 2 pages with peer-reviewed citations.
GAME AT MCDONALD’S Several years ago, McDonald’s ran a campaign in which it gave game cards to its customers. These game cards made it possible for customers to win hamburgers, french fries, soft drinks, and other fast-food items, as well as cash prizes. Each card had 10 covered spots that could be uncovered by rubbing them with a coin. Beneath three of these spots were “zaps.” Beneath the other seven spots were names of prizes, two of which were identical. For example, one card might have two pictures of a hamburger, one picture of a Coke, one of french fries, one of a milk shake, one of $5, one of $1000, and three zaps. For this card the customer could win a ham- burger. To win on any card, the customer had to uncover the two matching spots (which showed the potential prize for that
card) before uncovering a zap; any card with a zap uncovered was automatically void. Assuming that the two matches and the three zaps were arranged randomly on the cards, what is the probability of a customer winning?
We label the two matching spots M1 and M2, and the three zaps Z1, Z2, and Z3. Then the probability of winning is the probability of uncovering M1 and M2 before uncovering Z1, Z2, or Z3. In this case the relevant set of outcomes is the set of all orderings of M1, M2, Z1, Z2, and Z3, shown in the order they are uncovered. As far as the outcome of the game is concerned, the other five spots on the card are irrelevant. Thus, an outcome such as M2, M1, Z3, Z1, Z2 is a winner, whereas M2, Z2, Z1, M1, Z3 is a loser. Actually, the first of these would be declared a winner as soon as M1 was uncovered, and the second would be declared a loser as soon as Z2 was uncovered. However, we show the whole sequence of M’s and Z’s so that we can count outcomes correctly. We then find the probability of winning using an equally likely argument. Specifically, we divide the number of outcomes that are winners by the total number of outcomes. It can be shown that the number of out- comes that are winners is 12, whereas the total number of outcomes is 120. Therefore, the probability of a winner is 12/120 = 0.1.
This calculation, which shows that, on average, 1 out of 10 cards could be win- ners was obviously important for McDonald’s. Actually, this provides only an upper bound on the fraction of cards where a prize was awarded. Many customers threw their cards away without playing the game, and even some of the winners neglected to claim their prizes. So, for example, McDonald’s knew that if they made 50,000 cards where a milk shake was the winning prize, somewhat less than 5000 milk shakes would be given away. Knowing approximately what their expected “losses” would be from win- ning cards, McDonald’s was able to design the game (how many cards of each type to print) so that the expected extra revenue (from customers attracted to the game) would cover the expected losses.
CHAPTER 5 Probability and Probability Distributions
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5-1 Introduction 1 8 5
5-1 Introduction A key aspect of solving real business problems is dealing appropriately with uncertainty. This involves recognizing explicitly that uncertainty exists and using quantitative methods to model uncertainty. If you want to develop realistic business models, you cannot simply act as if uncertainty doesn’t exist. For example, if you don’t know next month’s demand, you shouldn’t build a model that assumes next month’s demand is a sure 1500 units. This is only wishful thinking. You should instead incorporate demand uncertainty explicitly into your model. To do this, you need to know how to deal quantitatively with uncertainty. This involves probability and probability distributions. We introduce these topics in this chapter and then use them in a number of later chapters.
There are many sources of uncertainty. Demands for products are uncertain, times between arrivals to a supermarket are uncertain, stock price returns are uncertain, changes in interest rates are uncertain, and so on. In many situations, the uncertain quantity— demand, time between arrivals, stock price return, change in interest rate—is a numerical quantity. In the language of probability, it is called a random variable. More formally, a random variable associates a numerical value with each possible random outcome.
Associated with each random variable is a probability distribution that lists all of the possible values of the random variable and their corresponding probabilities. A proba- bility distribution provides very useful information. It not only indicates the possible val- ues of the random variable but it also indicates how likely they are. For example, it is useful to know that the possible demands for a product are, say, 100, 200, 300, and 400, but it is even more useful to know that the probabilities of these four values are, say, 0.1, 0.2, 0.4, and 0.3. This implies, for example, that there is a 70% chance that demand will be at least 300.
It is often useful to summarize the information from a probability distribution with numerical summary measures. These include the mean, variance, and standard deviation. As their names imply, these summary measures are much like the corresponding summary measures in Chapters 2 and 3. However, they are not identical. The summary measures in this chapter are based on probability distributions, not an observed data set. We will use numerical examples to explain the difference between the two—and how they are related.
The purpose of this chapter is to explain the basic concepts and tools necessary to work with probability distributions and their summary measures. The chapter then dis- cusses several important probability distributions, particularly the normal distribution and the binomial distribution, in some detail.
Modeling uncertainty, as we will be doing in the next chapter and later in Chapters 15 and 16, is sometimes difficult, depending on the complexity of the model, and it is easy to get so caught up in the details that you lose sight of the big picture. For this rea- son, the flow chart in Figure 5.1 is useful. (A colored version of this chart is available in the file Modeling Uncertainty Flow Chart.xlsx.) Take a close look at the middle row of this chart. You begin with inputs, some of which are uncertain quantities, you use Excel® formulas to incorporate the logic of the model, and the result is probability distributions of important outputs. Finally, you use this information to make decisions. (The abbreviation EMV stands for expected monetary value. It is discussed extensively in Chapter 6.) The other boxes in the chart deal with implementation issues, particularly with the software you can use to perform the analysis. Study this chart carefully and return to it as you pro- ceed through the next few chapters and Chapters 15 and 16.
Before proceeding, we discuss two terms you often hear in the business world: uncertainty and risk. They are sometimes used interchangeably, but they are not really the same. You typically have no control over uncertainty; it is something that simply exists. A good example is the uncertainty in exchange rates. You cannot be sure what the exchange rate between the U.S. dollar and the euro will be a year from now. All you can try to do is measure this uncertainty with a probability distribution.
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1 8 6 C H A P T E R 5 P r o b a b i l i t y a n d P r o b a b i l i t y D i s t r i b u t i o n s
In contrast, risk depends on your position. Even though you don’t know what the exchange rate will be, it makes no difference to you—there is no risk to you—if you have no European investments, you aren’t planning a trip to Europe, and you don’t have to buy or sell anything in Europe. You have risk only when you stand to gain or lose money depending on the eventual exchange rate. Of course, the type of risk you face depends on your position. If you are holding euros in a money market account, you are hoping that euros gain value relative to the dollar. But if you are planning a European vacation, you are hoping that euros lose value relative to the dollar.
Uncertainty and risk are inherent in many of the examples in this book. By learning about probability, you will learn how to measure uncertainty, and you will also learn how to measure the risks involved in various decisions. One important topic you will not learn much about is risk mitigation by various types of hedging. For example, if you know you have to purchase a large quantity of some product from Europe a year from now, you face the risk that the value of the euro could increase dramatically, thus cost- ing you a lot of money. Fortunately, there are ways to hedge this risk, so that if the euro does increase relative to the dollar, your hedge minimizes your losses. Hedging risk is an extremely important topic, and it is practiced daily in the real world, but it is beyond the scope of this book.
5-2 Probability Essentials We begin with a brief discussion of probability. The concept of probability is one that you all encounter in everyday life. When a weather forecaster states that the chance of rain is 70%, he or she is making a probability statement. When a sports commentator states that the odds against the Golden State Warriors winning the NBA Championship are 3 to 1, he or she is also making a probability statement. The concept of probability is quite intuitive. However, the rules of probability are not always as intuitive or easy to master. We examine the most important of these rules in this section.
Assess probability distributions of uncertain inputs:
Decide which inputs are important for the model.
1. Which are known with certainty? 2. Which are uncertain?
Two fundamental approaches:
1. Build an exact probability model that incorporates the rules of probability. (Pros: It is exact and amenable to sensitivity analysis. Cons: It is often difficult mathematically, maybe not even possible.)
2. Build a simulation model. (Pros: It is typically much easier, especially with add-ins like DADM_Tools or @RISK, and extremely versatile. Cons: It is only approximate and runs can be time consuming for complex models).
For simulation models, random values for uncertain inputs are necessary.
1. They can sometimes be generated with built-in Excel functions. This often involves tricks and can be obscure.
2. Add-ins like DADM_Tools or @RISK provide functions that make it much easier.
Examine important outputs.
The result of these formulas should be probability distribution(s) of important output(s). Summarize these probability distributions with (1) histograms (risk profiles), (2) means and standard deviations, (3) percentiles, (4) possibly others.
Model the problem.
Use Excel formulas to relate inputs to important outputs, i.e., enter the business logic.
Make decisions based on this information.
Criterion is usually EMV, but it could be something else, e.g., minimize the probability of losing money.
1. If a lot of historical data is available, find the distribution that best fits the data.
2. Choose a probability distribution (normal? triangular?) that seems reasonable. Add-ins like DADM_Tools or @RISK are helpful for exploring distributions.
3. Gather relevant information, ask experts, and do the best you can.
This is an overview of spreadsheet modeling with uncertainty. The main process is in red. The blue boxes deal with implementation issues.
For simulation models, this can be done “manually” with data tables and built-in functions like AVERAGE, STDEV.S, etc. But add-ins like DADM_Tools or @RISK take care of these bookkeeping details automatically.
Use decision trees, made easier with add-in like DADM_Tools or PrecisionTree, if the number of possible decisions and the number of possible outcomes are not too large.
Figure 5.1 Flow Chart for Modeling Uncertainty
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5-2 Probability Essentials 1 8 7
As the examples in the preceding paragraph illustrate, probabilities are sometimes expressed as percentages or odds. However, these can easily be converted to probabilities on a 0-to-1 scale. If the chance of rain is 70%, then the probability of rain is 0.7. Similarly, if the odds against the Warriors winning are 3 to 1, then the probability of the Warriors winning is 1/4 (or 0.25).
There are only a few probability rules you need to know, and they are discussed in the next few subsections. Surprisingly, these are the only rules you need to know. Probability is not an easy topic, and a more thorough discussion of it would lead to considerable math- ematical complexity, well beyond the level of this book. However, it is all based on the few relatively simple rules discussed next.
5-2a Rule of Complements The simplest probability rule involves the complement of an event. If A is any event, then the complement of A, denoted by A (or in some books by Ac), is the event that A does not occur. For example, if A is the event that the Dow Jones Industrial Average will finish the year at or above the 25,000 mark, then the complement of A is that the Dow will finish the year below 25,000.
If the probability of A is P(A), then the probability of its complement, P(A), is given by Equation (5.1). Equivalently, the probability of an event and the probability of its com- plement sum to 1. For example, if you believe that the probability of the Dow finishing at or above 25,000 is 0.25, then the probability that it will finish the year below 25,000 is 1 − 0.25 = 0.75.
A probability is a number between 0 and 1 that measures the likelihood that some event will occur. An event with probability 0 cannot occur, whereas an event with probability 1 is certain to occur. An event with probability greater than 0 and less than 1 involves uncertainty. The closer its probability is to 1, the more likely it is to occur.
Rule of Complements
P1A2 5 1 2 P1A2 (5.1)
5-2b Addition Rule Events are mutually exclusive if at most one of them can occur. That is, if one of them occurs, then none of the others can occur. For example, consider the following three events involving a company’s annual revenue for the coming year: (1) revenue is less than $1 mil- lion, (2) revenue is at least $1 million but less than $2 million, and (3) revenue is at least $2 million. Clearly, only one of these events can occur. Therefore, they are mutually exclu- sive. They are also exhaustive events, which means that they exhaust all possibilities— one of these three events must occur. Let A1 through An be any n events. Then the addition rule of probability involves the probability that at least one of these events will occur. In general, this probability is quite complex, but it simplifies considerably when the events are mutually exclusive. In this case the probability that at least one of the events will occur is the sum of their individual probabilities, as shown in Equation (5.2). Of course, when the events are mutually exclusive, “at least one” is equivalent to “exactly one.” In addition,
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1 8 8 C H A P T E R 5 P r o b a b i l i t y a n d P r o b a b i l i t y D i s t r i b u t i o n s
In a typical application, the events A1 through An are chosen to partition the set of all possible outcomes into a number of mutually exclusive events. For example, in terms of a company’s annual revenue, define A1 as “revenue is less than $1 million,” A2 as “revenue is at least $1 million but less than $2 million,” and A3 as “revenue is at least $2 million.” Then these three events are mutually exclusive and exhaustive. Therefore, their probabilities must sum to 1. Suppose these probabilities are P(A1) = 0.5, P(A2) = 0.3, and P(A3) = 0.2. (Note that these probabilities do sum to 1.) Then the additive rule enables you to calculate other probabilities. For example, the event that revenue is at least $1 million is the event that either A2 or A3 occurs. From the addition rule, its prob- ability is
P1revenue is at least $1 million2 5 P1A22 1 P1A32 5 0.5
Similarly,
P1revenue is less than $2 million2 5 P1A12 1 P1A22 5 0.8
and
P1revenue is less than $1 million or at least $2 million2 5 P1A12 1 P1A32 5 0.7
Again, the addition rule works only for mutually exclusive events. If the events over- lap, the situation is more complex. For example, suppose you are dealt a bridge hand (13 cards from a 52-card deck). Let H, D, C, and S, respectively, be the events that you get at least 5 hearts, at least 5 diamonds, at least 5 clubs, and at least 5 spades. What is the probability that at least one of these four events occurs? It is not the sum of their individual probabilities because they are not mutually exclusive. For example, you could get a hand with 5 hearts and 5 spades. Probabilities such as this one are actually quite difficult to calculate, and we will not pursue them here. Just be aware that the addition rule does not apply unless the events are mutually exclusive.
5-2c Conditional Probability and the Multiplication Rule Probabilities are always assessed relative to the information currently available. As new information becomes available, probabilities can change. For example, if you read that Steph Curry suffered a season-ending injury, your assessment of the probability that the Warriors will win the NBA Championship would obviously change. (It would probably become 0!) A formal way to revise probabilities on the basis of new information is to use conditional probabilities.
Let A and B be any events with probabilities P(A) and P(B). Typically, the probability P(A) is assessed without knowledge of whether B occurs. However, if you are told that B has occurred, then the probability of A might change. The new probability of A is called the conditional probability of A given B, and it is denoted by P(A1B). Note that there is still uncertainty involving the event to the left of the vertical bar in this notation; you do not know whether it will occur. However, there is no uncertainty involving the event to the right of the vertical bar; you know that it has occurred. The conditional probability can be calculated with the following formula.
Addition Rule for Mutually Exclusive Events
P1at least one of A1 through An2 5 P1A12 1 P1A22 1 c 1 P1An2 (5.2)
if the events A1 through An are exhaustive, then the probability is 1 because one of the events is certain to occur.
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5-2 Probability Essentials 1 8 9
The numerator in this formula is the probability that both A and B occur. This proba- bility must be known to find P1AuB2. However, in some applications P1AuB2 and P(B) are known. Then you can multiply both sides of Equation (5.3) by P(B) to obtain the follow- ing multiplication rule for P(A and B).
Multiplication Rule
P1A and B2 5 P1AuB2 P1B2 (5.4)
The conditional probability formula and the multiplication rule are both valid; in fact, they are equivalent. The one you use depends on which probabilities you know and which you want to calculate, as illustrated in Example 5.1.
EXAMPLE
5.1 ASSESSING UNCERTAINTY AT BENDER COMPANY Bender Company supplies contractors with materials for the construction of houses. The company currently has a contract with one of its customers to fill an order by the end of July. However, there is some uncertainty about whether this deadline can be met, due to uncertainty about whether Bender will receive the materials it needs from one of its suppliers by the middle of July. Right now it is July 1. How can the uncertainty in this situation be assessed?
Objective To apply probability rules to calculate the probability that Bender will meet its end-of-July deadline, given the information the company has at the beginning of July.
Solution Let A be the event that Bender meets its end-of-July deadline, and let B be the event that Bender receives the materials from its supplier by the middle of July. The probabilities Bender is best able to assess on July 1 are probably P(B) and P1AuB2. At the beginning of July, Bender might estimate that the chances of getting the materials on time from its supplier are 2 out of 3, so that P(B) = 2/3. Also, thinking ahead, Bender estimates that if it receives the required materials on time, the chances of meeting the end-of-July deadline are 3 out of 4. This is a conditional probability statement that P1AuB2 5 3>4. Then the multiplication rule implies that
P1A and B2 5 P1AuB2P1B2 5 13>42 12>32 5 0.5
That is, there is a fifty–fifty chance that Bender will get its materials on time and meet its end-of-July deadline. This uncertain situation is depicted graphically in the form of a probability tree in Figure 5.2. Note that Bender ini-
tially faces (at the leftmost branch of the tree) the uncertainty of whether event B or its complement will occur. Regardless of whether event B occurs, Bender must next confront the uncertainty regarding event A. This uncertainty is reflected in the set of two pairs of branches in the right half of the tree. Hence, there are four mutually exclusive outcomes regarding the two uncer- tain events, as listed to the right of the tree. Initially, Bender is interested in the first possible outcome, the joint occurrence of
Conditional Probability
P1AuB2 5 P1A and B2
P1B2 (5.3)
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1 9 0 C H A P T E R 5 P r o b a b i l i t y a n d P r o b a b i l i t y D i s t r i b u t i o n s
events A and B, the top probability in the figure. Another way to compute this probability is to multiply the probabilities asso- ciated with the branches leading to this outcome, that is, the probability of B times the probability of A given B. As the figure indicates, this is 13/42 12/32, or 0.5.
P(A and B) = (3/4)(2/3)
P(A and B) = (1/4)(2/3)
P(A and B) = (1/5)(1/3)
P(A and B) = (4/5)(1/3)
Figure 5.2 Probability Tree for Bender Example
5-2d Probabilistic Independence A concept that is closely tied to conditional probability is probabilistic independence. You just saw that the probability of an event A can depend on whether another event B has occurred. Typically, the probabilities P(A), P1AuB2, and P1AuB2 are all different, as in Example 5.1. However, there are situations where all of these probabilities are equal. In this case, A and B are probabilistically independent events. This does not mean they are mutually exclusive. Rather, probabilistic independence means that knowledge of one event is of no value when assessing the probability of the other.
There are several other probabilities of interest in this example. First, let B be the complement of B; it is the event that the materials from the supplier do not arrive on time. We know that P1B2 5 1 2 P1B2 5 1>3 from the rule of complements. How- ever, we do not yet know the conditional probability P1AuB2, the probability that Bender will meet its end-of-July deadline, given that it does not receive the materials from the supplier on time. In particular, P1AuB2 is not equal to 1 2 P1AuB2. (Can you see why?) Suppose Bender estimates that the chances of meeting the end-of-July deadline are 1 out of 5 if the materials do not arrive on time, so that P1AuB2 5 1>5. Then a second use of the multiplication rule gives
P1A and B2 5 P1AuB2P1B2 5 11>52 11>32 5 0.0667
In words, there is only 1 chance out of 15 that the materials will not arrive on time and Bender will meet its end-of-July dead- line. This is the third (from the top) probability listed at the right of the tree.
The bottom line for Bender is whether it will meet its end-of-July deadline. After mid-July, this probability is either P1AuB2 5 3>4 or P1AuB2 5 1>5 because by this time, Bender will know whether the materials arrived on time. But on July 1, the relevant probability is P(A)—there is still uncertainty about whether B or B will occur. Fortunately, you can calculate P(A) from the probabilities you already know. The logic is that A consists of the two mutually exclusive events (A and B) and 1A and B2. That is, if A is to occur, it must occur with B or with B. Therefore, the addition rule for mutually exclusive events implies that
P1A2 5 P1A and B2 1 P1A and B2 5 1>2 1 1>15 5 17>30 5 0.5667
The chances are 17 out of 30 that Bender will meet its end-of-July deadline, given the information it has at the beginning of July.
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5-2 Probability Essentials 1 9 1
How can you tell whether events are probabilistically independent? Unfortunately, this issue usually cannot be settled with mathematical arguments. Typically, you nee
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