MABJ2
How do you use probability in your everyday life? Based on what you have
learned in this unit, will you make any changes? Why, or why not?
Your journal entry must be at least 200 words in length. No references or
citations are necessary.
LDR 5301, Methods of Analysis for Business Operations 1
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
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
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.
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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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LDR 5301, Methods of Analysis for Business Operations 2
UNIT x STUDY GUIDE
Title
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)
LDR 5301, Methods of Analysis for Business Operations 3
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Title
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).
LDR 5301, Methods of Analysis for Business Operations 4
UNIT x STUDY GUIDE
Title
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. Multiplechoice exams: Multiplechoice 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
multiplechoice 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 5050 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
LDR 5301, Methods of Analysis for Business Operations 5
UNIT x STUDY GUIDE
Title
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
LDR 5301, Methods of Analysis for Business Operations 6
UNIT x STUDY GUIDE
Title
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
LDR 5301, Methods of Analysis for Business Operations 7
UNIT x STUDY GUIDE
Title
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/personalfinance/110415/whydoeshouse
alwayswinlookcasinoprofitability.asp
Probability Formula. (n.d.). Relative frequency. https://probabilityformula.org/relativefrequency.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
LDR 5301, Methods of Analysis for Business Operations 8
UNIT x STUDY GUIDE
Title
Rice, D. (2018, October 16). Mega Millions jumps to $868 million, secondlargest jackpot in US history. USA
Today. https://www.usatoday.com/story/news/nation/2018/10/16/megamillionspowerballodds
winningjackpot/1656732002/
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.
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 SelfTest on page 55. You can use the key in the back of the book in
Appendix H to check your answers for SelfTests.
Finally, complete Problems 214, 216, 218 on page 56. You can use the answer key in Appendix G in the
back of the textbook in order to check your answers.
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Course Learning Outcomes for Unit II
Required Unit Resources
Unit Lesson
Introduction to Probability
The Fundamental Concepts of Probability
Types of Probability
Objective Approach
Subjective Approach
Relative Frequency
Probability Distributions: Expected Value
Conclusion
References
Suggested Unit Resources
Learning Activities (Nongraded)