Discrete Distribution (Playing Card Experiment), Discrete Random Variables | Statistics

Learning objectives

Discrete Distribution (Playing Card Experiment)

Student Learning Outcomes

The student will compare empirical data and a theoretical distribution to determine if an everyday experiment fits a discrete distribution.
The student will compare technology-generated simulation and a theoretical distribution.
The student will demonstrate an understanding of long-term probabilities.

Supplies

One full deck of playing cards
Programmable calculator

Procedure for Empirical Data

The experimental procedure for empirical data is to pick one card from a deck of shuffled cards.

The theoretical probability of picking a diamond from a deck is ________.
Shuffle a deck of cards.
Pick one card from it.
Record whether it was a diamond or not a diamond.
Put the card back and reshuffle.
Do this a total of 10 times.
Record the number of diamonds picked.
Let X = number of diamonds. Theoretically, X ~ B(_____,_____)

Procedure for Simulation

Repeat the experimental procedure using a programmable calculator.

Use the randInt function to generate data. Consider 1 to be spades, 2 to be hearts, 3 to be diamonds, and 4 to be clubs. Generate 10 draws of cards with four suits with randInt(1,4,10).
Let $X = number of diamonds$ . Theoretically, X ~ B(_____,_____).

Organize the Empirical Data

Record the number of diamonds picked for your class with playing cards in lab1_tbl001. Then calculate the relative frequency.

x	Frequency	Relative Frequency
0	__________	__________
1	__________	__________
2	__________	__________
3	__________	__________
4	__________	__________
5	__________	__________
6	__________	__________
7	__________	__________
8	__________	__________
9	__________	__________
10	__________	__________

Calculate the following:
1. $\bar{x}$ = ________
2. s = ________
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 diamonds. The y-axis is labeled Relative frequency.

Organize the Simulation Data

Use lab1_tbl002 to record the number of diamonds picked for your class using the calculator simulation. Calculate the relative frequency.

X	Frequency	Relative Frequency
0	__________	__________
1	__________	__________
2	__________	__________
3	__________	__________
4	__________	__________
5	__________	__________
6	__________	__________
7	__________	__________
8	__________	__________
9	__________	__________
10	__________	__________

Calculate the following:
1. $\bar{x}$ = ________
2. s = ________
Construct a histogram of the simulation 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 diamonds. The y-axis is labeled Relative frequency.

Theoretical Distribution

Build the theoretical PDF chart based on the distribution in the Procedure section.

x P(x)

0

1

2

3

4

5

6

7

8

9

10
Calculate the following:
1. μ = ____________
2. σ = ____________
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.

Using the Data

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

P(x = 3) = ________
P(1 < x < 4) = ________
P(x ≥ 8) = ________

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

RF(x = 3) = ________
RF(1 < x < 4) = ________
RF(x ≥ 8) = ________

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

RF(x = 3) = ________
RF(1 < x < 4) = ________
RF(x ≥ 8) = ________

Discussion Questions

For Questions 1 and 2, think about the shapes of the two 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, empirical, and simulation distributions. Use complete sentences.
Describe the three most significant differences between the graphs or distributions of the theoretical, empirical, and simulation distributions.
Using your answers from Questions 1 and 2, does it appear that the two sets of 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 lab1_tbl001, lab1_tbl002, or lab1_tbl003 to change, and how would it change? Why? Why wouldn’t the other table(s) change?

HOMEWORK

Use a programmable calculator to simulate a binomial distribution.

How would you use the randInt function to simulate the number of successes in five trials of an experiment with two outcomes, each of which has a .5 probability of occurring?
Use the randInt function to simulate 10 observations of the random variable in Part A.
Find the sample mean and sample standard deviation.
Compare the sample mean and sample standard deviation to the theoretical mean and the theoretical standard deviation.

You can use randInt (0,1,5) to generate five trials of the experiment. Count the number of 1s generated to determine the number of successes.
Student answers may vary.
Student answers may vary.
The theoretical mean is $(5) (.5) = 2.5$ . The theoretical standard deviation is $\sqrt{(5) (.5) (0.5)} = \sqrt{1.25}$ .

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.

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