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LDR 5301, Methods of Analysis for Business Operations 1

Course Learning Outcomes for Unit II Upon completion of this unit, students should be able to:

2. Distinguish between the approaches to determining probability. 2.1 Explain the probability and relative frequency of given data. 2.2 Compute relative frequency. 2.3 Prepare a report for a given problem using probability data.

Course/Unit

Learning Outcomes Learning Activity

2.1

Unit Lesson Chapter 2, pp. 21–26, 30–34, 51 Video Segment: Power of Prediction Video Segment: Chance and Percentage Video Segment: Making Accurate Predictions Video Segment: Search and Rescue Video Segment: Artificial Intelligence Unit II Assignment

2.2 Unit Lesson Chapter 2, pp. 21–26, 30–34, 51 Unit II Assignment

2.3

Unit Lesson Chapter 2, pp. 21–26, 30–34, 51 Video Segment: Power of Prediction Video Segment: Chance and Percentage Video Segment: Making Accurate Predictions Video Segment: Search and Rescue Video Segment: Artificial Intelligence Unit II Assignment

Required Unit Resources Chapter 2: Probability Concepts and Applications, pp. 21–26, 30–34, 51 In order to access the following resources, click the links below. Watch the following segments from the full video below: Power of Prediction (Segment 1 of 12), Chance and Percentage (Segment 3 of 12), Making Accurate Predictions (Segment 6 of 12), Search and Rescue (Segment 10 of 12), and Artificial Intelligence (Segment 11 of 12) PBS (Producer). (2018). Prediction by the numbers [Video]. Films on Demand.

https://libraryresources.columbiasouthern.edu/login?auth=CAS&url=https://fod.infobase.com/PortalPl aylists.aspx?wID=273866&xtid=169058

The transcript for these segments can be found by clicking on “Transcript” in the gray bar to the right of the video in the Films on Demand database.

UNIT II STUDY GUIDE Probability Concepts and Applications

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Unit Lesson

Introduction to Probability Think about how often you hear someone say, “What are the odds of that happening?” Or, “I would have a better chance of being hit by lightning or eaten by a shark than winning the Powerball lottery.” Figure 1 provides some insight into the odds and probability of some events.

Here is an even a better example: According to Gambling.net (n.d.), approximately 1.6 billion people gamble in a given year, and approximately 4.6 billion people have gambled in their lifetimes. Imagine saying this: “I’m going to Las Vegas, and I am going to win it big!” Remember, the casino is in business to earn a profit and provide shareholders with a return on their investment (equity, debt). With the business they are in, they must also carefully give a calculated payout to the customer. The casino wants you to win, but they also want you to return to spend more. The gaming industry is in business to earn profits (as is any business); however, every business survives by having you come back and spend more on the goods and services, or for the thrill of gambling. Maverick (2019) noted that the odds are in favor of the house winning on a large majority of the games that are played. Since the gaming industry’s goal is gross profit, it still must have a payout to continue to draw individuals to their casino. On some games, casinos only make a 1% to 2% profit, but on others, they could make up to 25%. Casinos are not out to bankrupt people, but they do want to earn money. In the game of roulette, for example, the house has a 5.25% edge; so, for every $1,000,000 played, the casino can expect to make around $50,000. That means that the customers are taking home the remaining $950,000. As you can see, probability is a large part of the casinos’ strategy. So, what is probability? According to Render et al. (2018), “a probability is a numerical statement about the likelihood that an event will occur” (p. 21). We have experienced probability on a daily basis in our lives with regard to forecasting, project management, game theory, and, of course, the weather. Regarding forecasting, probability is involved because forecasting uses data to predict future behaviors such as selling a product (cars, fashion, etc.). Forecasters award a probability that is either high or low (aggressive or conservative) depending on their assumptions. Of course, there is risk in this analysis. Imagine that retailers see in the Farmer’s Almanac that it will be a very cold winter in the Northeastern United States and across the Midwest. These retailers in the Northeast order extra Canada goose jackets and hats, only to be rewarded with a total bust in the weather forecast where a warm current comes up from the south and makes the winter season more like summer. Just the opposite could happen too, if in a predicted warm winter, retailers cut back on sweaters, gloves, jackets, boots, and then a cold front comes down from Canada and the small winter clothes inventory is immediately depleted. Again, what are the odds and probability of this happening?

Figure 1: Odds and probability of events (Rice, 2018)

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Looking at project management, contracts are made and fees based on completion dates. What hurts contractors and major building projects is the weather. The weather is a forecast, again, a best estimate based on data that a certain event will happen. The probability of finishing a project during monsoon season in a specific area would be very low, and the completion of a building in subzero weather would also have a low probability. Consequently, the contract would have to be adjusted with either a longer completion date or a reduction in contract pricing.

The Fundamental Concepts of Probability Let’s look at the fundamental concepts of probability. There are two basic rules you should take away from this lesson. The first is:

This can be expressed with this formula:

Can you think of a probability, given this rule that will never occur? Remember, a probability of 1 means it is always expected to occur; although the probability is 1 out of 300,000,000, there is still the probability it will occur. One example of this scenario can be seen occasionally on ESPN during college football season when they begin to rank the teams (top 10, top 25) and provide percentages, which is also the probability of winning teams moving up to a playoff position (top four). This is especially true when a team ranked seventh has a chance if the number 10 team beats the number one ranked team, and the number 12 team beats the number two team, and so on. The second rule can be expressed this way:

Types of Probability

Really? There are types of probability? Yes! There are two ways to determine probability:

• Objective Approach • Subjective Approach

The probability, P, of any event or state of nature occurring is greater than or equal to 0 and less than or equal to 1. A probability of 0 indicates that an equal event is never expected to occur. A probability of 1 means than an event is always expected to occur (Render et al., 2018, p. 22).

0 < P(event) < 1

The sum of the simple probabilities for all possible outcomes of an activity must equal 1. Regardless of how many probabilities are determined, they must adhere to these two rules (Render et al., 2018, p. 22).

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Objective Approach As individuals, we know what both of these words (objective and subjective) mean, especially when taking examinations. The objective probability approach deals with a single, clear, definite answer. There is perfect objectivity in the score because there is only one correct answer. These are two examples of an objective approach:

1. Multiple-choice exams: Multiple-choice exams are very objective, meaning they have a single correct answer. When taking these exams, you actually have a probability or percentage of selecting the correct answer. On a multiple choice test with four to five answers, only one is right. Therefore, you have a 20 to 25% chance of selecting the correct answer if you do not know it. As one college professor once said, “You did not have to study for this exam. I am giving you the answers on the multiple-choice exam. The answer is right in front of you on each question; you just have to select it.”

2. True/false exams: The same goes for true/false tests; you have a 50-50 probability of selecting the correct answer.

Subjective Approach

Render et al. (2018) noted that the subjective approach to probability looks at individual experiences and judgments when making the estimates. This usually occurs when logic and past history are not available for evaluation. The experiences of individuals weigh heavily on the possible events occurring. There is an old joke that represents this subjectivity approach: Put 20 attorneys in a room and give them a case to solve with an outcome. As a result, you will probably get 20 different approaches and solutions. Can you think of an example of subjective probability? Here are some examples:

• What is the probability of the stock market crashing? • What is the probability of interest rates going negative? • What is the probability of the Unite States defaulting on its debt? • What is the probability of a worldwide pandemic? • What is the probability of an epidemic in your town, city, or state? • What is the probability of you being president of a major automotive company next year?

Relative Frequency

A final takeaway from types of probability is the relative frequency approach;

P (event) = Number of occurrences of the event Total Number of trials or outcomes

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The example below demonstrates the concept of relative frequency. It is a small restaurant’s information regarding the types of steaks sold during Monday night dinner service:

Type of Steak at Restaurant

Number Frequency Selected

Probability = 1.00

Rare

1 1/25 = .04

Medium Rare

12 12/25 = .48

Medium

10 10/25 = .40

Well Done

2 2/25 = .08

Total 25 1.00 Also let’s look at a problem dealing with relative frequency. Twelve cards are numbered from one to 12 and none are repeated. One of the 12 cards is drawn at a time, the number is recorded, and the card is then put back in the deck. This is repeated 50 times and the following chart shows what was observed (Probability formula, n.d.):

4 10 6 12 5 10 5 6 12 11 1 6 3 1 6 12 1 5 4 4 12 6 6 1 12 4 10 12 3 8 6 12 9 8 4 3 8 12 3 12 5 4 11 12 5 5 5 8 5 12

(Probability Formula, n.d.) We will now try to find the probably of 12. This is how we would solve this. We see that occurrence of 12 is 11 times in all 50 trials. So, in this case, the probability of occurrence of 12 is simply the relative frequency of 12. Using the formula that was introduced earlier, we get these results: P (occurrence of 12) = 11/50 = 0.22 Relative frequency = 0.22 = 22% So in an experiment of 50 trials, 12 has occurred 22%. Let’s look at another problem. In a class of 42 students, there are three modes of transportation to school. Twenty students travel by school bus, 15 travel by car, and the remaining seven students walk. Let’s find the relative frequencies (Probability Formula, n.d.). To solve this, we would again use our relative frequency formula. Remember, the total number of students is 42, and the following list details their transportation modes.

• Number of students riding the school bus = 20 • Number of students arriving by car = 15 • Number of students who walk = 7

Let’s now do the math. The relative frequencies are:

• School bus = 20/42 = 0.48 • Car = 15/42 = 0.36 • Walk = 7/42 = 0.17

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Probability Distributions: Expected Value Let’s take a look at the expected value of a probability distribution. What is the expected value? According to Render et al. (2018), the expected value (EV) of a probability distribution is the anticipated mean value of the computed values.

Let’s have some fun with an example of the expected value formula because it is a little intimidating. Do you remember the TV game show Deal or No Deal? The game consisted of one contestant and 26 models displaying 26 briefcases. Each briefcase held a numerical value from $0.01 (one cent) to $1 million. The contestant started the game by selecting one out of the 26 suitcases. Think about this: What is the probability here of selecting the $1 million case? It is 1/26 = .038, which can be rounded up to 4%. This is not optimal. As the game progresses, the contestant picks two or three briefcases each round. As the numbers are revealed, the banker begins to compute the odds—the probability—of the chances of the contestant having a $1 million case, and the banker begins to offer the contestant payout money to quit the game (based on the expected value of the remaining cases). So, let’s assume you are the contestant, and you reached the point where you have four briefcases remaining: The briefcases are: $25, $1,000, $250,000, and $1,000,000 (one of which is yours but you do not know what your case contains). Being a smart contestant who has taken this Master of Science in Leadership class, you are hoping for a commercial break so you can take out your pocket calculator and determine the probability and expected value of the event and determine if the banker’s offer is good. You remember this: The two basic rules of probability state that the possible outcome must equal

Look at the table below. You will notice that:

1. You have a 25% chance of picking any briefcase with one choice. 2. The expected payout is 25% times that value. 3. The overall expected payout (or expected monetary value [EVM]) from the banker should be the total

of all expected payouts: $312, 756.

EX=∑P(Xi)×Xi

1.0 < P(event) ,< 1

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BRIEFCASE VALUE

EXPECTED PAYOUT

$25

$25 x .25 $6.25

$1,000

$1,000 x .25 $250

$250,000

$250,000 x .25 $62,500

$1,000,000

$1,000,000 x .25 $250,000

$312,756 expected monetary value (EMV)

Based on these calculations, $312,756 is the amount we would expect the banker to offer to buy our briefcase. After the commercial break, the game show host comes back and tells you the banker offers you the following: a $200,000 payout. Here are some things to consider.

• Will you take it? Why? You already know the $200,000 is below the EMV. • Assume you continue to play and you narrow down the two cases to $250,000 and $1 million. You

have a 50/50 chance of having the $1 million at that point. o What would be the expected payout from the banker? o Go ahead and do the math with EMV (as in the above table) o Do you take the payout (assuming the banker has figured out you are one smart individual and

provides the exact EMV), or do you go for the $1M knowing you can walk away with $200K, which is below the EMV?

These would all be things to consider, and application of the concepts regarding probability that we have discussed in this unit.

Conclusion This lesson is all about decisions and risk. This lesson discussed the fundamentals of probability and how to calculate it. We then put probability into perspective with some examples; it is fundamentally based on numbers and calculation with odds. As we have seen in the diagram at the beginning of the lesson, you have a higher chance of being eaten by a shark than winning the lottery. You have also been exposed to how game shows use probability and expected values, and then display odds that are in the house’s favor to create excitement.

References Gambling.net. (n.d.). Gambling statistics: Gambling stats from around the world.

https://www.gambling.net/statistics.php Maverick, J. B. (2019, October 29). Why does the house always win? A look at casino profitability.

Investopedia. https://www.investopedia.com/articles/personal-finance/110415/why-does-house- always-win-look-casino-profitability.asp

Probability Formula. (n.d.). Relative frequency. https://probabilityformula.org/relative-frequency.html Render, B., Stair, R. M., Jr., Hanna, M. E., & Hale, T. S. (2018). Quantitative analysis for management (13th

ed.). Pearson. https://online.vitalsource.com/#/books/9780134518558

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Rice, D. (2018, October 16). Mega Millions jumps to $868 million, second-largest jackpot in US history. USA Today. https://www.usatoday.com/story/news/nation/2018/10/16/mega-millions-powerball-odds- winning-jackpot/1656732002/

Suggested Unit Resources In order to access the following resources, click the links below. The Chapter 2 PowerPoint Presentation will summarize and reinforce the information from this chapter in your textbook. You can also view a PDF of the Chapter 2 presentation. Learning Activities (Nongraded) Nongraded Learning Activities are provided to aid students in their course of study. You do not have to submit them. If you have questions, contact your instructor for further guidance and information. For an overview of the chapter equations, review the Key Equations on page 51 of the textbook. Then, complete question 15 on the Self-Test on page 55. You can use the key in the back of the book in Appendix H to check your answers for Self-Tests. Finally, complete Problems 2-14, 2-16, 2-18 on page 56. You can use the answer key in Appendix G in the back of the textbook in order to check your answers.