Section 1: Probability Basics

Introduction/Welcome to the class

During this class you’ll learn what probability tells you, how to avoid the fallacies that mislead people, how to calculate probability by breaking the event down, what conditional probability tells you, how to use Bayes’ Rule to incorporate your knowledge and how to use expected value for your best guess on the average outcome. No math is required beyond basic arithmetic. Spreadsheets are used to demonstrate randomness and formulas, but their use is optional.

Probability: What and Why: Expected and Unexpected Values

Probability is both a science and a precise way to talk about likelihood. You’ll learn to convert odds to probability and calculate expected value. You’ll learn to break a problem up into equally likely outcomes to easily compute probabilities. You’ll watch some simulated coin flips and random draws to develop intuition about randomness.

Probability and the Real World/It’s not Just Count and Divide

Many of the real-world problems you confront are too complex for count and divide. In fact, subjectivity often enters into probability, so that people may legitimately assess different probability values for the same event based on their knowledge. We broaden our definition of probability to help you solve the problems you need to solve.

Section 2: Probability Fallacies and How to Avoid Them

Common Fallacies/Simple but Prevalent Misuses of Probability

Don’t get sucked in by these common fallacies that lead to bad decision-making. Probability is deceptively simple, so beware.

Key Fallacies/Spurious Patterns and Confusion of the Inverse

One of the most common fallacies is seeing randomly produced patterns and thinking they mean something, leading to spurious conclusions. The other is confusion of the inverse, in which conditional probabilities get turned around, giving spurious implications, leading to often silly conclusions.

Section 3: Conditional Probabilities

Conditional Probabilities/When the Situation or Group Being Considered Makes a Difference

Conditional probability is useful when you’re interested in events happening one after another, but it’s broader than that. It also helps you understand the confusion of the inverse and the concepts of false negative and false positive.

False Positives and False Negatives/How Good is the Prediction Really?

If you understand this, you’re ahead of most people, including many doctors. A prediction can fail in two different ways, a false positive or a false negative. We look at the implication of these values if you get a positive test for a rare condition. Beware the confusion of the inverse.

COVID Example and Summary of False Positives and False Negatives

We look at an example for COVID tests. We summarize the four possible true and false outcomes of a test.

Section 4: Calculating Probabilities

Calculating Simple Probabilities/Equally Likely Outcomes and Historical Data

You can apply the count-and-divide technique to historical data you’ve collected. We look at an example of a bank finding a group more likely to default on their loans.

Combining Probabilities to Calculate More Complex Ones

The real-world events you’re faced with usually have complex probabilities depending on multiple things happening. Learn how to combine simple probabilities to get the answers.

Try It/Examples of the Techniques You’ve Learned

Try out these techniques in some exercises with solutions.

Section 5: Bayes’ Rule

Bayes’ Rule/Enriching Your Probabilities with Outside Knowledge

Bayes’ Rule is a powerful way for you to incorporate knowledge about the outside world to get more realistic conditional probabilities. See an example in which the raw statistics seem to indicate one thing, but Bayes’ Rule and information about the total population gives a totally different answer.

Using Bayes’ Rule/Solving Confusion of the Inverse and False Positives

Bayes’ Rule allows you to solve the confusion of the inverse by calculating the reversed conditional. It also helps to clear up a lot of confusion involved in false negatives and false positives.

Section 6: Expected Value

Expected Value/How Much Can You Expect on Average

The expected value tells you what you can expect on average. It also helps you to compare two alternatives under uncertainty. See an example of expected value used to compare alternative strategies for a popular game.

Section 7: The Wrap-Up

Challenges/The Wedding Challenge and the Birthday Challenge

Test your probability intuition. Here are two problems with surprising answers. Yes, the solutions are included.

The Monty Hall Problem

This is a famous puzzle that makes people crazy. Even very smart people don’t believe the answer. Can you see it? 

Summary/Key Concepts and Formulas

This is a quick review of all you’re learned in the class. There’s also a cheat sheet in the resources of the Introduction that summarizes all this on one page.