Skip to ContentGo to accessibility page
OpenStax Logo
Statistics

3.6 Probability Topics

Statistics3.6 Probability Topics

3.6 Probability Topics

Stats Lab

Probability Topics

Student Learning Outcomes

  • The student will use theoretical and empirical methods to estimate probabilities.
  • The student will appraise the differences between the two estimates.
  • The student will demonstrate an understanding of long-term relative frequencies.

Do the ExperimentCount out 40 mixed-color candies, which is approximately one small bag’s worth. Record the number of each color in Table 3.12. Use the information from this table to complete Table 3.13. Next, put the candies in a cup. The experiment is to pick two candies, one at a time. Do not look into the cup as you pick them. The first time through, replace the first candy before picking the second one. Record the results in the With Replacement column of Table 3.14. Do this 24 times. The second time through, after picking the first candy, do not replace it before picking the second one. Then, pick the second one. Record the results in the Without Replacement column section of Table 3.15. After you record the pick, put both candies back. Do this a total of 24 times, also. Use the data from Table 3.15 to calculate the empirical probability questions. Leave your answers in unreduced fractional form. Do not multiply out any fractions.

Color Quantity
Yellow (Y)
Green (G)
Blue (BL)
Brown (B)
Orange (O)
Red (R)
Table 3.12 Population
With Replacement Without Replacement
P(2 reds)
P(R1B2 OR B1R2)
P(R1 AND G2)
P(G2|R1)
P(no yellows)
P(doubles)
P(no doubles)
Table 3.13 Theoretical Probabilities

Note

G2 = green on second pick, R1 = red on first pick, B1 = brown on first pick, B2 = brown on second pick,
doubles = both picks are the same color.

With Replacement Without Replacement
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
( __ , __ ) ( __ , __ ) ( __ , __ ) ( __ , __ )
Table 3.14 Empirical Results
With Replacement Without Replacement
P(2 reds)
P(R1B2 OR B1R2)
P(R1 AND G2)
P(G2|R1)
P(no yellows)
P(doubles)
P(no doubles)
Table 3.15 Empirical Probabilities

Discussion Questions

  1. Why are the With Replacement and Without Replacement probabilities different?
  2. Convert P(no yellows) to decimal format for both Theoretical With Replacement and for Empirical With Replacement. Round to four decimal places.
    1. Theoretical With Replacement: P(no yellows) = _______
    2. Empirical With Replacement: P(no yellows) = _______
    3. Are the decimal values close? Did you expect them to be closer together or farther apart? Why?
  3. If you increased the number of times you picked two candies to 240 times, why would empirical probability values change?
  4. Would this change (see Question 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know?
  5. Explain the differences in what P(G1 AND R2) and P(R1|G2) represent. Hint: Think about the sample space for each probability.
PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's prior written permission.
  • This book uses the Creative Commons Attribution License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute Texas Education Agency (TEA), and must distribute any derivative works under the same license. The original material is available at: https://www.texasgateway.org/book/tea-statistics . Changes were made to the original material, including updates to art, structure, and other content updates.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/statistics/pages/1-introduction

  • If you are redistributing all or part of this book in a noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/statistics/pages/1-introduction

Citation information

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 Texas Education Agency (TEA). The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.