Posted: March 12th, 2023

MAB_J3

LDR 5301, Methods of Analysis for Business Operations 1

Course Learning Outcomes for Unit III

Upon completion of this unit, students should be able to:

2. Distinguish between the approaches to determining probability.
2.1 Examine different types of probability distribution.
2.2 Analyze how different approaches to determining probability can be used in the real world.

3. Contrast the major differences between the normal distribution and the exponential and Poisson

distributions.
3.1 Summarize a real-world example of normal distribution or Poisson distribution.
3.2 Describe how normal or Poisson distribution was used in a scenario.
3.3 Explain why either normal or Poisson distribution was used over another type of distribution.

Course/Unit

Learning Outcomes

Learning Activity

2.1

Unit Lesson

Chapter 2, pp. 32–34, 38–42, 47–51
Video Segment: Graphs: Normal Distributions
Unit III Article Review

2.2

Unit Lesson
Chapter 2, pp. 32–34, 38–42, 47–51
Video Segment: Graphs: Normal Distributions
Unit III Article Review

3.1

Unit Lesson
Chapter 2, pp. 32–34, 38–42, 47–51
Video Segment: Graphs: Normal Distributions
Unit III Article Review

3.2

Unit Lesson
Chapter 2, pp. 32–34, 38–42, 47–51
Video Segment: Graphs: Normal Distributions
Unit III Article Review

3.3

Unit Lesson
Chapter 2, pp. 32–34, 38–42, 47–51
Video Segment: Graphs: Normal Distributions
Unit III Article Review

Required Unit Resources

Chapter 2: Probability Concepts and Applications, pp. 32–34, 38–42, 47–51

In order to access the following resource, click the link below.

Uniview Worldwide (Producer). (2005). Graphs: Normal distributions (Segment 6 of 9) [Video]. In Organizing

quantitative data. Films on Demand.
https://libraryresources.columbiasouthern.edu/login?auth=CAS&url=https://fod.infobase.com/PortalPl
aylists.aspx?wID=273866&xtid=36200&loid=43047

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

UNIT III STUDY GUIDE
Probability Distributions:
Part 1

https://libraryresources.columbiasouthern.edu/login?auth=CAS&url=https://fod.infobase.com/PortalPlaylists.aspx?wID=273866&xtid=36200&loid=43047

LDR 5301, Methods of Analysis for Business Operations 2

UNIT x STUDY GUIDE
Title

Unit Lesson

Introduction

You have now entered the third unit of the course. You should begin to feel comfortable with the content and
delivery of the unit lessons and the assignments. The biggest takeaway each week is not that you can crunch
numbers, but that you can comprehend, analyze, and evaluate what the numbers mean in the context of the
problem or written assignment presented. You can think at a higher level of what the data mean. Remember,
the data do not solve the problem. What solves the problem is applying the data to a rational, well-thought-out
decision and plan.

In this lesson, we examine the probability of distributions. We will look at the normal distribution, the
exponential distribution, and the Poisson distribution. As we look at the formulas and the shapes of the
curves, the most important part is comprehending how these distributions apply. You need to have a basic
understanding of what the formulas look like and how they work; however, in the real world, all of this is done
by an algorithm computer. This means that you, the user, just plug the numbers into the software, and the
algorithm does the work for you. Again, what is key is that you interpret and comprehend the data, and know
how to explain the numbers within your given situation.

Let’s start this out with a quick review.

The distributions follow three main rules we have already covered:

• the events are mutually exclusive and collectively exhaustive;
• the individual probability values are between 0 and 1 inclusive; and
• the total of the probability values add up to 1 (Render et al., 2018).

Render et al. (2018) provide examples of distributions in Figures 2.7 (probability distributions), 2.8 (normal
distribution) 2.16 (exponential distribution), and 2.18 (Poisson distributions) on page 38. Take a look at these
figures so that you get an idea of how each distribution is shaped.

Random Variables

What is a random variable? A random variable is a variable that “assigns a real number to every possible
outcome or event in an experiment” (Render et al., 2018, p. 30). An example of this is the outcome of flipping

a coin (heads or tails, or one out
of two) or rolling a die (which has
six sides, or one out of six). Now,
a probability distribution is the
set of all possible values of a
random variable and their
associated probabilities (Render
et al., 2018)

Example of Complex Probability Problem

A card collector has an inventory of 40 cards, and he is playing a game with his friend. He has the following
teams that are both from the Eastern Conference and in the Atlantic Division:

• 10 Toronto Maple Leaf Topps Cards (TMLT),
• 15 Toronto Maple Leaf Upper Deck Cards (TMLUD),
• 3 Boston Bruins Upper Deck Cards (BBUD), and
• 12 Boston Bruins Topps Cards (BBT).

His friend randomly selects a card from the pile, and it is an Upper Deck card. Note: The Upper Deck Cards
are worth three times the price value of any Topps card.

Probability distribution of rolling a die.

LDR 5301, Methods of Analysis for Business Operations 3

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Title

What is the probability that the card is also a Boston Bruin?

1. Compute the probabilities given the information.
2. What if the question was: What is the probability that the card is a Toronto Maple Leaf?

Answer:

First: Your first step is to compute the probability for each card over the entire lot (lot of cards = 40)

P(TMLT) = 10/40 = .25
P(TMLUD)= 15/40 = .37
P(BBT)= 12/40 = .30
P(BBUD)= 3/40 = .075

Next, you want to find the probability for the select Toronto Cards (A) that are Topps and Upper Deck. Then,
do the same for Boston (B). The reason for doing this is because you can pick either of the four types as
noted.

A: P(TML) = P(TMLT) + P(TMLUD) = (.25) + (.37) = .62

B: P(BB) = P (BBT) + P (BBUD) = (.30) + (.075) = 375

Next, you need to compute the probability of just Upper Deck (C), so you can see what the probability is of
just selecting that type.

C: P(UD) = P(TMLUD) + P (BBUD) = (.37) + (.075) = .445

Now, compute the same as above but for Topps only (D).

D: P(T) = P(TMLT) + P (BBT) = (.25) + (.30) = .55

Now, it is time to complete your final step and computation to find out the answer to the first question: What is
the probability of the card being a Boston Bruin card (E)?

E: P(UD/BB) =.075, which are pretty slim odds of picking that type of card

What would it be for a Toronto Maple Leaf UD?

P(UD/TML) = .37

Now, let’s move onto looking at each of the distributions and see where and how they are used in real-world
applications. We will discuss two distributions in this lesson, the normal and Poisson distributions.

Probability Distributions

Normal Distribution

The normal distribution is a symmetric probability distribution that, when graphed, presents itself as a bell
curve. This means that data near the mean happens more often than data that is farther from the mean
(Chen, 2019).

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Title

Why is it called a normal distribution? Look at its shape; it is normal and equal on both sides of the middle (u),
which is the average. To use this distribution model, you need the mean (average of the numbers) and the
standard deviation (the square root of the variance). In easier terms, it is the measurement of the group that is
spread out from the mean or expected value. When looking at a normal distribution, you see it is shaped like
a bell curve (narrow on the ends, tall in the middle) When the standard deviation becomes smaller, the curve
becomes flatter; however, it is just the opposite when it increases and the curve becomes taller. Render et al.
(2018) do a good job of displaying this with Figure 2.8 in the textbook. As the standard deviation changes, you
can see how the curve changes shape. In a normal distribution, both sides of the mean must equal each
other.

The normal distribution is likely to be seen in our daily lives through the following examples:

• a professor assigning grades based on a bell curve,
• rush hour times for your city,
• the average pizza delivery times for your local area, and
• LA Fitness’s popular operating times, as evidenced by customer attendance.

So, why does this matter? Well, by knowing this distribution, there are a few things that can be taken away.

• If you were the manager of this LA Fitness, you would not want your cleaning crews or maintenance
crews working during peak hours.

• If you were a customer, you might rethink your exercise time based on peak attendance hours. Why?
Because each gym has limited resources based on the area’s population size. There are a limited
number of treadmills, elliptical machines, stair steppers, and, in the weight room, weight machines.

You may still be thinking you will never use this information, but you use it every day and do not realize you
are using it to make decisions. You are making decisions based on probability, mathematical analysis, and
distributions (we will cover decision-making in a future unit). The takeaway here is that by using quantitative
analysis, distributions, and probability, you give yourself a higher probability of success in your decision-
making. That is what the data’s ground zero does for you as a leader, manager, or CEO/executive; it
improves your success rate.

Normal distribution
(Render et al., 2018)

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UNIT x STUDY GUIDE
Title

Poisson Distribution

What is the Poisson distribution, and where is it used? The key point in using this distribution is that it
“describes situations in which customers arrive independently during a certain time interval, and the number
of arrivals depends on the length of time of the interval” (Render et al., 2018, p. 48). Let’s look at the formula
for the Poisson distribution:

𝑃𝑃(𝑋𝑋) =
𝜆𝜆𝑥𝑥𝑒𝑒−𝜆𝜆

𝑋𝑋!

Where:

P(X) = probability of exactly X arrivals or occurrences

= average number of arrivals per unit of time (the mean arrival rate)

e = 2.178, the base of the natural algorithm

X = number of occurrences (0, 1, 2…)

Let’s have some fun now exploring where the Poisson distribution is used.

Recall again what a Poisson distribution measures (as stated in a previous paragraph). Take a look at the
examples below, and see if you can see this applied:

• the hospital emergency room,
• customers arriving at a bank window,
• passengers arriving at an airport, or
• telephone calls to a central exchange (Render et al., 2018)

Another great example is accidents at traffic signals in Chicago. This is another Poisson distribution that is
totally random in nature. No one knows when a traffic accident is going to happen, but looking at this annual
data, we can see the trend is down (see following charts). This actual study was looking at the 380 red light
cameras that were used at 190 intersections (Shah, 2014). After reading the article, we find that the individual
conducting the study attributed the rise and fall to two reasons: drivers being negligent in driving and drivers
being more aware of their driving (Shah, 2014). Although the article is general in nature, what really could be

Sample Poisson Distributions
(Render et al., 2018)

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UNIT x STUDY GUIDE
Title

attained here is breaking down the data by month, as well as looking at weather patterns, construction,
speeding, and if tickets received were a deterrent.

Distribution Perspectives

Jerzy Letkowski (n.d.) from Western New England University did a study on the applications of the Poisson
probability distribution. His findings were rather interesting. Review the following list of some of the
applications of the Poisson distribution, and think about what the distribution might look like:

• number of times a strand of DNA mutates in a specific time frame (Wikipedia-Poisson, 2012 as cited
in Letkowski, 2012);

• amount of bankruptcies per month (Jaggia, Kelly, 2012 p.158 as cited in Letkowski, 2012);
• number of cars at a car wash each hour (Anderson et al., 2012, p. 236 as cited in Letkowski, 2012);
• number of times a computer network fails each day (Levine, 2010, p. 197 as cited in Letkowski,

2012);
• birth and death rate each year or month (Weiers, 2008, p. 187 as cited in Letkowski, 2012); and
• number of people who call a store’s customer service department in a given time frame (Donnelly, Jr.,

2012, p. 215 as cited in Letkowski, 2012).

Can you think of a Poisson distribution of data in your life you could plot? One example might be the number
of FedEx and UPS trucks on the road that you see when you go on a trip.

Conclusion

In this unit, we looked at probability concepts and applications. Most important was the outlining of the three
rules that help us frame a distribution problem. A probability example was provided that had multiple factors.
As was noted, the probabilities were assigned to each entity and then computed with an overall probability of
selection. We then discussed distributions (normal, Poisson) through real-world examples. Each example was
displayed with a graph so you could see how the data is reflected in the distribution. The examples displayed
for each distribution are particularly interesting because, as individuals, we see them occurring every day
without knowing what they really mean from a data standpoint.

References

Chen, J. (2020, March 31). Normal distribution. Investopedia.

https://www.investopedia.com/terms/n/normaldistribution.asp

19,500

20,000

20,500

21,000

21,500

2009 2010 2011 2012

Accidents at Chicago Traffic
Signals

24.50%

25.00%

25.50%

26.00%

26.50%

27.00%

2009 2010 2011 2012

Percentage of Traffic
Accidents

(Adapted from Eyeing Chicago, 2014)

LDR 5301, Methods of Analysis for Business Operations 7

UNIT x STUDY GUIDE
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Letkowski J. (2012, March 22-24). Applications of the Poisson probability distribution [Conference session],
Academic and Business Research Institute Conference, San Antonio, TX, United States.
http://www.aabri.com/SA12Manuscripts/SA12083

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

Shah, R. (2014, August 31). Red light cameras in Chicago: Following surveillance in the second city. Eyeing

Chicago. http://eyeingchicago.com/red-light-camera-study/

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, read the Key Equations on page 51 of the textbook.

Then, complete solved problems 2-1, 2-2, and 2-3 on page 52 and Self-Test problem 1-15 on pages 54–55.
You can use the key in the back of the book in Appendix H to check your answers for Self-Tests.

For the Solved Problems, the problem is presented first, followed by its solution. Challenge yourself to apply
what you have learned, and see if you can work out the problems without first looking at the solution. Only
use the solution to check your own work.

Course Learning Outcomes for Unit III

Learning Activity

Required Unit Resources

Unit Lesson

Introduction

Random Variables

Example of Complex Probability Problem

Probability Distributions

Normal Distribution

Poisson Distribution

Distribution Perspectives

Conclusion

References

Learning Activities (Nongraded)

MAB_J3

Explain how you take the two basic laws of probability into account in your everyday life, and give real-world examples of each (other than flipping a coin and rolling a die).

Your journal entry must be at least 200 words in length. No references or citations are necessary.