Jari Lipponen of the Silverhaven Home for Abandoned Miniature Horses (SHAMH) needs funds to maintain operations. He can either apply for a government grant or run a local fundraiser, but the demands on his time are too high for him to be able to do both.
Jari believes he has a 90% chance that he will win a grant of $25,000 if he submits the grant application. He estimates that a local fundraiser, he has a 30% chance of raising $30,000 and 70% chance of raising $20,000. The EMV of the fundraiser option is $23,000, higher than $22,500, the EMV of applying for the grant. Based on the initial analysis, Jari should organize a fundraiser.
Jari has expressed uncertainty about his estimate for the probabilities of the two levels of fundraising success. How low would the probability of the raising $30,000 have to be to change his decision?
For Jari to change his initial decision, the EMV of the fundraising option would have to be less than $22,500. That would be the case if the probability of the high fundraising success — valued at $30,000 — were less than 25%.
Jari needs $20,000 to operate the SHAMH through the next year. Raising $25,000 or more would permit Jari to operate the SHAMH and expand the capacity of the operation, thereby allowing even more abandoned
miniature horses to be saved. This year will be Jari's last running the SHAMH, and he really wants to expand its capacity before he leaves.
If he's unable to raise at least $25,000 to cover the SHAMH expansion, he'll be very disappointed. How highly would Jari have to value his disappointment in order for him to prefer the government grant option that is more likely to secure funds for expansion? Assume that his original probability assessments for the success of the fundraiser are correct.
a. Less than $833.33.
This is not the best answer. Think about what range of values of the disappointment cost will lead to an EMV for the grant-writing scenario that is higher than the EMV for the fundraiser scenario.
b. Greater than $833.33. This is the best answer. c. Less than $714.29.
This is not the best answer. Think about which scenarios will not bring in enough funds to expand the SHAMH.
d. Greater than $714.29.
This is not the best answer. Think about which scenarios will not bring in enough funds to expand the SHAMH.
If Jari is only moderately successful in his fundraising activities, he won't be able to expand the SHAMH on his watch. The disappointment cost D should be subtracted from the outcome value for the scenario in which he takes in only $20,000 in contributions. After incorporating the disappointment cost, the EMV of the fundraiser option becomes $23,000 - 0.7D.
However, if he applies for the grant and doesn't win it, he won't be able to expand, either. The disappointment cost D must also be subtracted from the outcome value in the scenario in which he writes the grant but doesn't win it. After incorporating the disappointment cost, the EMV of the grant-writing option becomes $22,500 - 0.1D.
The breakeven value is the value of disappointment for which the EMVs of the two options are equal. Thus, to find the breakeven value of D, we must set up an inequality between the two EMVs and solve for D. In this case, the disappointment cost would have to be greater than $833.33 in order for Jari to prefer the grant- writing option.
Decision Analysis II
Conditional Probabilities
After a Kahana breakfast of crab benedict, you meet once more with Leo. Dining Chez Tethys: The Market Research Problem
So, I've been thinking: my decision is heavily dependent on the likelihood that the Chez Tethys will be a success. Couldn't we do some market research to find out how interested our target market would be? Then I could make a better decision ! one that would be less risky!
Sure, Leo. But market research costs money. How much would you be willing to pay for information that would help you better predict the success of the Chez Tethys?
Hmmm. Good question...frankly, I don't have a clue how to even start thinking about that. Can you two help me? "Whatever market research Leo has in mind," remarks Alice, "it won't reveal with certainty how consumers will take to the Tethys. In other words, we're uncertain about the Tethys' chance of success, and we'll be uncertain that the market research we collect is accurate."
Joint and Marginal Probabilities
Pondering two layers of uncertainty you begin to feel vertigo...
When analyzing a decision, we typically use estimates for the probabilities and financial implications of various outcomes that could occur. Often, we can imagine additional information — scientific tests, market research data, or professional expertise — that would help make our estimates more accurate. How much should we be willing to pay for this type of information? And how do we incorporate new information into our decision analysis? Before we answer these questions, we'll need to expand our understanding of probability and introduce the important concepts of conditional probability and statistical independence. Let's look at an example first.
British automaker Chariot's most sought-after model is the Ben Hur. Consumers love the Bennie, as it's affectionately called. It was offered as a limited edition: to date, only 1,000 Bennies are on the road in Britain. We'll take a closer look at two properties of the Bennie: its "Color" and its "Stereo."
The Bennie comes in two color options: Burgundy and Champagne. Also, the Bennie is offered with a high-end car stereo system by Sweetone. Let's look at a table of the 1,000 Bennies currently on the road and see how the two properties — "Color" and "Stereo" — are distributed.
Of the 1,000 Bennies, 150 are Burgundy and have a Sweetone stereo. 600 are Burgundy and do not have the Sweetone stereo. Furthermore, there are 50 Champagne Bennies with the stereo and 200 without.
Let's add another column to our table and fill in the total numbers of Burgundy and
Champagne Bennies. To calculate these numbers, we simply take the sums of the rows. For example, the number of Burgundy Bennies is simply the number of Burgundy Bennies with the Sweetone stereo added to the number of Burgundy Bennies without.
Next, in the final row, we'll fill in the total numbers of Bennies with and without the Sweetone stereo. Here, we simply take the sums of the two columns. In the bottom right cell, we enter the total number of cars: 1,000.
This number — 1,000 — should be equal to the sum of the numbers in the final row: the total number of Bennies with the Sweetone stereo added to the total number without one. Also, it should be equal to the sum of the numbers in the final column: the number of Burgundy Bennies plus the number of Champagne Bennies. Finally, since 1,000 is the total number of all
Bennies, it should be the sum of all the original numbers we entered into the table.
A Venn diagram is a useful graphical way to represent the contents of the table. The Burgundy rectangle on top represents the set of all Burgundy Bennies. The Champagne rectangle below represents the set of all Champagne Bennies. The rectangles do not intersect because a Bennie cannot be both Champagne and Burgundy.
We now add a patterned rectangle on the left to represent the set of Bennies with the Sweetone stereo. The area without the pattern represents the set of Bennies that are not equipped with the Sweetone. These areas intersect the rectangles that represent the
distribution of "color": some Burgundy Bennies are Sweetone-equipped; some are not. Some Champagne Bennies are Sweetone-equipped; some are not.
The sizes of the areas in the Venn diagram are directly proportional to the incidence of the different Bennie properties: Burgundy/Champagne and Sweetone/no Sweetone. We use Venn diagrams to effectively communicate information about sets of things — for example,
Burgundy cars and Sweetone-equipped cars — and their interactions.
The table is a useful tool for calculating proportions of potential interest to managers. For instance, we can find the proportion of Burgundy Bennies with the Sweetone stereo simply by locating the cell that contains their number and dividing it by the total number of cars: 15%. Or, to find the proportion of Burgundy Bennies in general, we find the cell that contains the total number of Burgundy Bennies and divide it by the total number of cars: 75%.
In this way we can create an entire table of proportions. These proportions can be interpreted as probabilities. For example, since 15% of the Bennies on the road are Burgundy-colored and Sweetone-equipped, the probability that a randomly selected Bennie will be Burgundy-colored and have the Sweetone is 15%.
The probability that a randomly selected Bennie will be Burgundy is 75%. Going forward, in talking about the table, we'll use the words "proportion" and "probability" interchangeably. The probabilities on the inside of the table are called joint probabilities: the probabilities of a single car having two particular Bennie features, for example, Burgundy-colored and
Sweetone-equipped. We'll denote joint probabilities in the following way: P(Burgundy & Sweetone) is the probability that a Bennie is Burgundy-colored and Sweetone-equipped. The probabilities of each "Color" or "Stereo" option occurring in a given Bennie — Burgundy or Champagne, Sweetone-equipped or not Sweetone-equipped — are often called marginal
probabilities. They are denoted simply as P(Burgundy), P(Champagne), P(Sweetone), and P(no Sweetone).
Information about the distribution of properties in populations is often available in terms of probabilities, so the table of probabilities is a very natural way to represent the Bennie data. Summary
For two events A and B with outcomes A1, A2, etc. and B1, B2, etc., respectively, the joint probability P(A1 & B1) is the probability that the uncertain event A has outcome A1 and the uncertain event B has outcome B1. The joint probabilities of all possible outcomes of two uncertain events can be summarized in a probability table. The marginal probability of the outcome A1 of the first uncertain event is the sum of the joint probabilities of outcomes A1 and all possible outcomes B1, B2, etc. of the second uncertain event.
Conditional Probabilities
Automaker Chariot's limited edition Ben Hur model comes in two possible colors — Champagne or Burgundy — and with or without a high-end Sweetone stereo. The table below shows the distribution of these properties — "Color" and "Stereo" — in the population of 1,000 Bennies that Chariot has sold to date.
Restricting our focus to the Burgundy Bennies, we ask the following question: among the set of Burgundy Bennies, what is the proportion of Burgundy Bennies with a Sweetone stereo?
Stated differently: what is the probability that a randomly selected Bennie has a Sweetone given that we know it is Burgundy?
a. 15%
This is not the best answer. Compare the total number of Burgundy Bennies that are Sweetone-equipped with the total number of Burgundy Bennies.
b. 20%
This is the best answer. c. 25%
This is not the best answer. Compare the total number of Burgundy Bennies that are Sweetone-equipped with the total number of Burgundy Bennies.
d. 75%
This is not the best answer. Compare the total number of Burgundy Bennies that are Sweetone-equipped with the total number of Burgundy Bennies.
To answer the question, we find the ratio of Sweetone-equipped Burgundy Bennies among the set of all Burgundy Bennies. That is, we divide the number of Bennies that are both Burgundy and have a Sweetone — 150 — by the total number of Bennies that are Burgundy — 750. The probability is 20%.
This probability is called a conditional probability: the probability that a Bennie is Sweetone- equipped given that it is Burgundy. We'll denote this probability P(Sweetone | Burgundy), and read the vertical line as "given."
We can calculate P(Sweetone | Champagne) as:
a. The number of Sweetone-equipped Bennies divided by the total number of Bennies.
This is not the best answer. Think about what a conditional probability represents.
b. The number of Sweetone-equipped Bennies divided by the number of Champagne Bennies.
This is not the best answer. Think about what a conditional probability represents.
c. The number of Champagne and Sweetone-equipped Bennies divided by the total number of Bennies.
This is not the best answer. Think about what a conditional probability represents.
d. The number of Champagne and Sweetone-equipped Bennies divided by the number of Champagne Bennies.
This is the best answer.
P(Sweetone | Champagne) is the number of Bennies that are Champagne and Sweetone- equipped divided by the total number of Champagne Bennies.
What is P(No Sweetone | Burgundy)?
Enter the percentage as a decimal number with two digits to the right of the decimal point (e.g., enter "50%" as "0.50"). Round if necessary.
Using the table of actual numbers of cars we can calculate P(No Sweetone | Burgundy) and a fourth conditional probability, P(No Sweetone | Champagne).
Earlier, we used the table of actual numbers of cars to calculate a table of probabilities. When given information as a table of probabilities, we can use the probabilities to calculate conditional probabilities as well: we simply form the ratios of the appropriate probabilities.
For example, the probability that a Bennie is Sweetone-equipped given that it is Burgundy is the probability of a Sweetone-equipped, Burgundy Bennie divided by the probability of a Burgundy Bennie.
In fact, a conditional probability is formally defined in terms of the ratio of a joint probability to a marginal probability.
If P(Sweetone | Burgundy) represents the probability that a Bennie is Sweetone-equipped given that it is Burgundy, what does P(Burgundy | Sweetone) represent?
a. The probability that a Bennie is Sweetone-equipped given that it is Burgundy.
This is not the best answer. Think about the different ratios you can form in the table of probabilities. b. The probability that a Bennie is Sweetone-equipped.
This is not the best answer. Think about the different ratios you can form in the table of probabilities. c. The probability that a Bennie is Burgundy given that it is Sweetone-equipped.
This is the best answer.
d. The probability that a Bennie is Burgundy.
This is not the best answer. Think about the different ratios you can form in the table of probabilities. P(Burgundy | Sweetone) represents the probability that a Bennie is Burgundy given that it is Sweetone-equipped. Note that P(Sweetone | Burgundy) and P(Burgundy | Sweetone) are not the same.
We can calculate the other conditional probabilities, this time conditioning the property "Color" on the
property "Stereo." It is useful to write the conditional probailities next to the table of probabilities to facilitate calculation: since P(Burgundy | Sweetone) and P(Champagne | Sweetone) require only the probabilities in the Sweetone column, we write them directly below the Sweetone column.
Similarly, since P(Burgundy | No Sweetone) and P(Champagne | No Sweetone) require only the probabilities in the "No Sweetone" column, we write them directly below the "No Sweetone" column. Note that the new rows mimic the rows in the original table: Burgundy on top and Champagne below it.
Similarly, we write the conditional probabilities for the property "Stereo" conditioned on the property "Color" to the right of the table. Again, we mimic the columns in the original table: a column for "Sweetone" and one for "No Sweetone." We now have a full table of joint, marginal, and conditional probabilities.
It is important to double check which event we are conditioning on, and to do a "reality check" to make sure our calculations are realistic. For example, the probability that a randomly chosen Indian citizen is the prime minister is nearly zero, but the probability that the prime minister of India is an Indian citizen is 100%! The table of joint probabilities informs our understanding of the likelihood of different properties or events in a crucial way: as we've seen in the Bennie example, once we have the joint probabilities, we can compute any conditional probability and any marginal probability.
Thus, when presented with a decision problem in which the outcomes are influenced by multiple uncertain events, constructing the joint probabilities of these events is almost always a wise first step.
Summary
The conditional probability P(A | B) is the probability of the outcome A of one uncertain event, given that the outcome B of a second uncertain event has already occurred. The table of joint probabilities provides all the information needed to compute all conditional probabilities. First calculate the marginal probabilities for each event, then compute the conditional probabilities as shown below:
Statistical Independence
Let's return once more to our Chariot Ben Hur example. The Bennie comes in two possible colors — Champagne or Burgundy — and with or without a high-end stereo by Sweetone. The probability table below shows the distribution of these properties - "Color" and "Stereo" - in the population of 1,000 Bennies that Chariot has sold to date.
same as the proportion of Sweetone-equipped Bennies in the overall population: 20%. In the language of conditional probabilities, this is the same as saying that P(Sweetone | Burgundy) = P(Sweetone).
In other words, if we randomly select a Bennie, then discovering that it is Burgundy gives us no additional information about whether or not it is equipped with a Sweetone stereo,
beyond what we had before we knew its color: we still think there is a 20% chance it has a Sweetone.
Similarly, the proportion of Burgundy Bennies in the Sweetone-equipped population is the same as the proportion of Burgundy Bennies in the overall population: P(Burgundy | Sweetone) = P(Burgundy) = 75%.
A Bennie's color tells us nothing about its stereo system. Its stereo system tells us nothing about its color. When this is true, we say that that the Bennie properties "Color" and
"Stereo" are statistically independent, or, more simply, independent.
In general, we can interpret the fact that two uncertain events are independent in the following way: knowing that one event has occurred gives us no additional information about whether or not the other event has. For example, the results of two spins of a wheel of fortune are independent. The first result does not reveal anything about the second.
We can confirm the independence of the Bennie's stereo and color by looking at our Venn diagram and noting that Sweetone-equipped Bennies occupy the same percentage in the population of burgundy Bennies — 20% — as they do in the entire Bennie population. Thus, to find the joint probability that a Bennie has a Sweetone stereo and is burgundy, we take 20% of the 75% of Bennies that are burgundy, giving us a joint probability of 15%. This property is true for any two statistically independent properties: the joint probabilities are simply the products of the marginal probabilities.
Although it may seem plausible to assume that certain properties are independent, managers who take statistical independence for granted do so at their peril. We need to verify the assumption that the properties are independent by looking at and evaluating data or by proving independence on the theoretical level. Statistical Dependence
When are two outcomes statistically dependent? Let's look at another optional feature of the Bennie: a unique factory-installed theft discouragement system (TDS).
Using Chariot's data, we can create the following table of the distribution of the properties "Stereo" and