Discrete Distribution (Lucky Dice Experiment), Discrete Random Variables | Statistics

Learning objectives

Discrete Distribution (Lucky Dice Experiment)

Student Learning Outcomes

The student will compare empirical data and a theoretical distribution to determine if a Tet gambling game fits a discrete distribution.
The student will demonstrate an understanding of long-term probabilities.

Supplies

One “Lucky Dice” game or three regular dice
One programming calculator

Procedure

Round answers to relative frequency and probability problems to four decimal places.

The experimental procedure is to bet on one object. Then, roll three Lucky Dice and count the number of matches. The number of matches will decide your profit.
What is the theoretical probability of one die matching the object?
Choose one object to place a bet on. Roll the three Lucky Dice. Count the number of matches.
Let X = number of matches. Theoretically, X ~ B(______,______)
Let Y = profit per game.

Organize the Data

In lab2_tbl001, fill in the y-value that corresponds to each x-value. Next, record the number of matches picked for your class. Then, calculate the relative frequency.

Complete the table.

x y Frequency Relative Frequency

0

1

2

3
Calculate the following:
1. $\bar{x}$ = _______
2. s_x = ________
3. $\bar{y}$ = _______
4. s_y = _______
Explain what $\bar{x}$ represents.
Explain what $\bar{y}$ represents.
Based upon the experiment, answer the following questions:
1. What was the average profit per game?
2. Did this represent an average win or loss per game?
3. How do you know? Answer in complete sentences.
Construct a histogram of the empirical data.
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a blank graph template. The x-axis is labeled Number of matches. The y-axis is labeled Relative frequency.

Theoretical Distribution

Build the theoretical PDF chart for x and y based on the distribution from the Procedure section.

x y P(x) = P(y)

0

1

2

3
Calculate the following:
1. μ_x = ________
2. σ_x = ________
3. μ_x = ________
Explain what μ_x represents.
Explain what μ_y represents.
Based upon theory, answer the following questions:
1. What was the expected profit per game?
2. Did the expected profit represent an average win or loss per game?
3. How do you know? Answer in complete sentences.
Construct a histogram of the theoretical distribution.
View original figure (external; image not reproduced in this text edition)
Original figure description: This is a blank graph template. The x-axis is labeled Number of diamonds. The y-axis is labeled Probability.

Use the Data

Use the data from the Theoretical Distribution section to calculate the following answers. Round your answers to four decimal places.

P(x = 3) = ________
P(0 < x < 3) = ________
P(x ≥ 2) = ________

Use the data from the Organize the Data section to calculate the following answers. Round your answers to four decimal places.

RF(x = 3) = ________
RF(0 < x < 3) = ________
RF(x ≥ 2) = ________

Discussion Question

For Questions 1 and 2, consider the graphs, the probabilities, the relative frequencies, the means, and the standard deviations.

Knowing that data vary, describe three similarities between the graphs and distributions of the theoretical and empirical distributions. Use complete sentences.
Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions.
Thinking about your answers to Questions 1 and 2, does it appear that the data fit the theoretical distribution? In complete sentences, explain why or why not.
Suppose that the experiment had been repeated 500 times. Would you expect lab2_tbl001 or lab2_tbl002 to change, and how would it change? Why? Why wouldn’t the other table change?

Attribution. Statistics, OpenStax, Rice University, and the contributing authors credited in the book. Licensed under Creative Commons Attribution 4.0 International. Source revision 7dea80ae28 of openstax/osbooks-statistics (2026-04-23). CC BY 4.0. Original source ↗. Text edition of a pinned CC BY 4.0 source revision of the OpenStax book, converted from CNXML by the EvidenceHelix importer. Original prose, learning objectives, tables, exercises, glossary and captions are preserved; images are not copied (figures link to the original source and keep their credits). No new content was added. Later revisions of this book can carry different licence terms and are not used.

x	y	Frequency	Relative Frequency
0
1
2
3

x	y	P(x) = P(y)
0
1
2
3