Lesson 2.2: Conditional Probability
Lesson Learning Objectives
- Explore conditional probabilities with a contingency table.
- Calculate probabilities using the General Multiplication Rule.
- Use probability trees to calculate conditional probabilities.
- Use Bayes Theorem to invert conditional probabilities.
Lesson 2.2 Checklist
| Learning activity | Graded? | Estimated time |
|---|---|---|
| Read OpenIntro Statistics section 3.2 and supplementary notes | No | 30 mins |
| Watch instructional video | No | 15 mins |
| Answer two lesson check-in questions | Yes | 15 mins |
| Work through virtual statistical software lab | No | 45 mins |
| Answer two virtual statistical software lab questions | Yes | 15 mins |
| Work on practice exercises | No | 1.5 hours |
| Explore suggested websites | No | 15 mins |
Learning Activities
Readings 📖 and Instructional Video 🎦
Defining Probability
Read Section 3.2: Conditional Probability in OpenIntro Statistics (Diez et al., 2019) CC BY-SA 3.0.
In Lesson 1.4: Summarizing Categorical Data, we learned how to summarize two categorical variables using a contingency table. With contingency tables, we are often interested in whether the variables are related. For example, does the pattern of frequency counts across the levels of one categorical variable change according to the level of the other categorical variable? Conditional probability provides a structure for understanding such a relationship. As you read, look up new terminology in the Glossary and self-assess your understanding by attempting the guided practice exercises.
Probability trees provide another way to understand conditional probability. Watch the video, Probability Trees (Çetinkaya-Rundel, 2013), about this intuitive way to solve conditional probability problems (duration 00:08:23).
Probability Trees and Conditional Probability
Read Supplementary Notes 2.2 to see more examples of probability trees and conditional probability.
Lesson Check-in Questions ✍
Virtual Statistical Software Lab 💻
Work through the virtual statistical software lab: Software Lab 2.2: Conditional Probability. This lab uses a compelling sports example to reinforce the ideas of independent events and conditional probability. You’ll also learn how to use computer code written in the language R to extend the capabilities of jamovi. You are not expected to write code for this course. Rather, this enhancement allows you to run any code that is supplied to you.
As you work through the lab, answer the exercises in the shaded boxes. These exercises are not graded, but the solutions are available: Software Lab 2.2 Solutions. The lab should take you no more than 45 minutes to complete.
Virtual Statistical Software Lab Questions ✍
Practice Exercises 🖊
Work on the following exercises in OpenIntro Statistics: Exercises 3.13, 3.15, 3.17, 3.19, and 3.21 and Chapter Exercise 3.41 (Diez et al., 2019) CC BY-SA 3.0. Check your answers using these solutions (Diez et al., 2019) CC BY-SA 3.0. You’ll deepen your understanding much more effectively if you genuinely attempt the questions by yourself before checking the solutions.
Work on the WeBWorK exercises, which are linked from your Moodle course. Check your answers using the solutions provided.
Suggested Websites 🌎
- Wrapping your head around conditional probability problems can be difficult, so check-out this Conditional Probability [Application] (MathIsFun, n.d.) lesson to work through another explanation.
- Brilliant is an online education website that features problems and courses in mathematics, physics, quantitative finance, and computer science. To access the site, you will need to create a free account. Some content requires a paid subscription.
- The strong selection of courses and quizzes will help you practice probability (Brilliant, n.d.).
- Do you think it is possible that one baseball player could have a higher batting average than another player in two consecutive years, but the second player has a higher average over the combined two year period? Surprisingly, that happened in both 1995 and 1996. David Justice’s batting average was higher than Derek Jeter’s, yet Jeter’s average was higher over the two year period. Find-out how in Simpson’s Paradox (Koswara et al., n.d.). If you don’t wish to create a Brilliant account, try Simpson’s Paradox and Interpreting Data (Grigg, 2018).
Media Attributions
Hütten – Kodak BW400CN, by Ric Capucho (2015), on Flickr, CC BY 2.0
References
Brilliant. (n.d.). Probability. https://brilliant.org/discrete-mathematics/
Capucho, R. (2015). Hütten – Kodak BW400CN [Photograph]. Flickr. https://flic.kr/p/qkfHHj
Çetinkaya-Rundel, M. (2013, Jan. 9). Probability trees [Video]. YouTube. https://www.youtube.com/watch?v=HxEz4ZHUY5Y
Diez, D. M., Çetinkaya-Rundel, M., Barr, C. D. (2019). OpenIntro Statistics (4th ed.). OpenIntro. https://www.openintro.org/book/os/
Grigg, T. (2018, Dec. 9). Simpson’s paradox and interpreting data. Towards Data Science. https://towardsdatascience.com/simpsons-paradox-and-interpreting-data-6a0443516765
Koswara, I., Aanlya, A., Goh, P. H., Ellinor, A., Bhagyanath, V., Ross, E., Khim, J., & Lin, C. (n.d.). Simpson’s paradox. In Brilliant. Retrieved July 10, 2023, from https://brilliant.org/wiki/simpsons-paradox/
MathIsFun. (n.d.). Conditional probability. https://www.mathsisfun.com/data/probability-events-conditional.html
