Week 10
Modelling Uncertainty: Probability
Business
⦿ Monday: Simple probability and tree diagrams
- Introduction to probability Pencast 6:54 | PDF
- Uncertainty surrounds us. Probability gives us the tools to model it. First we need to recognize the language that tips us off to the presence of uncertainty and particularly of probabilities: "chance", "odds" and of course "probability" itself. These terms all refer to fractions. This implies that probabilities will have always have decimal values between 0 and 1, and that the total probability of an event happening or not happening must be 1.0.
- Simple Probabilities Pencast 7:51 | PDF
- We begin with "simple" probability, i.e. the probability of discrete or atomic events, and consider coin tosses, dice rolls and dealing cards.
- Example: Two dice Pencast 8:51 | PDF
- Rolling two dice presents us with our first example of probability subtlety, emphasizes the importance of identifying equiprobable events, and provides our first model for compound events: a table of possibilities.
- Example: Coin tossing Pencast 12:21 | PDF
- Tossing a coin once is not the interesting, but what about the outcomes of multiple tosses? A second modelling tool, tree diagrams, help us map out the possibilities.
- Example: Basketball free throws Pencast 6:05 | PDF
- How many points can we expect our 69% free throw shooter to get in a one-and-one?
- Example: The playoffs Pencast 7:46 | PDF
- One team wins 3 of 4 regular season meetings with another. What is the probability they win a best of 3 playoff series?
- Tree algebra Pencast 5:05 | PDF
- We've been working with tree diagrams intuitively. Let's formalize the rules.
- The Chevalier de Méré's question Pencast 10:13 | PDF
- We've finally accumulated the tools necessary to answer the Chevalier de Méré's question. Was he right?
Exercises
If you're comfortable with the topics above you're ready to work on:
Supplementary resources:
In Introductory Statistics from
OpenStax
Chapter 3 covers this week's probability topics.
Chapter 3 in College Mathematics for Everyday life by Inigo, Jameson, Kozak, Lanzetta, and Sonier. It's a simpler presentation, but sadly doesn't have a table of contents for the chapter so you have to scroll to find what you want.
⦿ Wednesday: Conditional probability and Bayes' theorem
- Independence and conditional probability Pencast 9:55 | PDF
- The nice thing about coins and dice is that they don't know what happened on the previous toss or roll, but people and lots of systems do. That means we sometimes have to work with conditional instead of independent probabilities.
- Example: Colour blindness Pencast 11:03 | PDF
- Colour blindness is not evenly distributed across the sexes, which affects the odds of men and women being colour blind.
- Bayes' Theorem Pencast 4:25 | PDF
- Often we have a posterior condition, like a test result, and want to know what it tells us about the probability of a prior condition, like the presence of a disease.
- Example: Tuberculosis screening Pencast 11:39 | PDF
- You are randomly selected to be tested for tuberculosis. The test result is positive. Should you be worried?
- Example: That beetle! Pencast 11:12 | PDF
- You find a beetle with rare markings. Does that make it a rare beetle?
Exercises
If you're comfortable with the topics above you're ready to work on:
Supplementary resources:
In Introductory Statistics from
OpenStax
Chapter 3 covers this week's probability topics.
Chapter 3 in College Mathematics for Everyday life by Inigo, Jameson, Kozak, Lanzetta, and Sonier. It's a simpler presentation, but sadly doesn't have a table of contents for the chapter so you have to scroll to find what you want.
⦿ Friday: Probabilities from data
Where do probabilities come from? Some probabilities are primarly theoretical, like the chance of getting heads in a toss of a fair coin. But many have their origin in data. Sometimes that data is nicely presented, but when it isn't we need to do some organizing.
- Example: Grade 4 boys' sports participation Pencast 16:19 | PDF
- Some boys play soccer, some play baseball, and some play both. How can we organize the numbers?
- Example: Cancer statistics Pencast 11:02 | PDF
- Smoking, living in urban areas, and having a hazardous occupation are all risk factors for lung cancer. Let's break down the numbers.
Exercises
If you're comfortable with the topics above you're ready to work on:
Supplementary resources:
The only resource I found on the web that focusses just on using set theory for organizing data is this collection of short videos and notes from Ms. Chor at GHCI: Lesson 1: Organized Counting with Venn Diagrams (5.1). If anyone knows of, or finds, something better please share!
Assignment
Assignment 9 is posted. Due by 1 pm Friday March 15. Note that the problems are worth different numbers of marks. Use your time wisely.
- You can hand in work done by hand to me in class or in dropbox A19.
- Excel or other binary files should be sent as file attachments to an email message.
- Google sheets can be submitted by sharing a link to them.
- Work done in Desmos can be submitted by sharing a link to your saved graph.
Name the file you submit A9_firstname.pdf, e.g.
A9_Tim.pdf
If you have to submit multiple files, add a suffix number to each, e.g.
A9_Tim_1.pdf
A9_Tim_2.pdf