• In recent years, online gambling has become highly popular among young people. This unfortunately includes high school students, who sometimes lie about their age to gain access to apps for sports betting and prediction markets. This lesson presents an opportunity to connect the negative expected value of casino games to the negative expected value of many other types of gambling. Here’s a set of questions that could be posed to students: Now that we’ve calculated the expected value of roulette, why do you think casinos are profitable? Similarly, why do you think online sports betting companies and prediction markets are profitable?
• Consider introducing the idea of a “fair game” and asking students what they think this means. You can help guide them to the conclusion that a fair game has an expected value of zero, and then contrast this with casino games.
• All that said, if teenagers feel that they’re being “given a lecturing” about gambling, they may not fully internalize how harmful it can be. Instead, it can be more effective to provide an initially neutral framing and allow students to mathematically discover the negative expected value of casino games. This helps students feel ownership over the realization that gambling isn’t a consistent path to fortune.
• The notation P(X = x) can be challenging for students, especially in understanding the distinction between the capital and lower case letters. Some helpful framing: The capital letter defines a variable with possible outcomes determined by chance (e.g., “Let X represent the number of…”), while the corresponding lowercase letter represents a specific outcome among those possibilities.

First, download this lesson's Handout Key and read through its Discussion Question section. Then, check out our model discussion norms and the additional background notes below.

• First formalized in this 1979 paper by Daniel Kahneman and Amos Tversky, the concept of loss aversion applies directly to the Discussion Question. Loss aversion is the idea that “losses loom larger than gains.” As an example, the frustration of losing $10 is often greater than the happiness of finding a $10 bill on the street. Although loss aversion is sometimes framed as a cognitive bias, it can also be rational. In terms of the game proposed by the Discussion Question, losing $300,000 would harm most peoples’ lives more than winning $1,000,000 would help them. Winning $1,000,000 would afford someone more luxuries, which is great. But losing $300,000 could rob someone of their basic necessities, life stability, and future prospects (due to debt) – all of which would be devastating. So, in this case, loss aversion is rational, and the variance of this game is more pertinent than its expected value.
• To help support their understanding of variation and risk, ask students what would be better: betting $1,000 on 1 roulette spin, or betting $100 each time on 10 spins? Are they the same? Guide them to see that, while the expected value may be the same, the variability is different. In particular, the probability of losing the maximum amount ($1,000) is much higher in the first scenario. This can lead naturally into a discussion of risk tolerance.

• For simplicity, this lesson uses a European roulette wheel with one green space. Most American roulette wheels include two green spaces (0 and 00), which worsens the player’s expected value even further.
• Instructors can ask students whether people ever “beat the house,” and if so, how. Encourage students to consider the roles of luck or access to information (cheating). This connects to the stock market context used in lesson 2.A.5 and the idea of trading with insider information.
• An unfortunate reality is that some teenagers lie about their age in order to download betting apps and gamble online. In addition, teens may be particularly vulnerable to gambling addictions. Consider sharing gambling addiction resources with your classes, in case there may be some students whose eyes are opened by this lesson and who realize they may need help. Teachers can share the following resources:
  • The website of the National Council on Problem Gambling, which provides education and support related to problem gambling, including for adolescents.
  • The National Problem Gambling Helpline (1-800-522-4700), which provides confidential support and connects individuals to local resources.
  • For additional context, organizations such as the Child Mind Institute offer accessible explanations of online gambling risks and why teens may be particularly vulnerable.

• Emphasize that interpretations of expected value should include language such as “over many repetitions” or “after many trials.” This is a common place where students lose points on the AP exam. In addition, it can be helpful to emphasize that expected value does not need to be a specific possible outcome and should not be rounded to a whole number.
• To further emphasize the above point, instructors can point out that losing $2.80 (the expected value calculated in the lesson) is not a possible outcome for a single $100 bet on roulette. The only possible outcomes are winning $100 or losing $100. Instead, the expected value is average loss per play over many trials of roulette.
• Prompt students to recall that the Greek letter μ represents the mean of a population. Then ask why the same symbol is used for expected value. This highlights the important idea that a probability distribution represents all possible outcomes and their probabilities.