What Do You Expect to Happen When…?

What do expect to happen when the weather report calls for an 80% chance of rain for tomorrow? What do you expect to happen when your favorite major league player, who has a batting average of .300, comes to bat, or when you buy a raffle ticket for $1 in an attempt to win a prize worth $800? These questions can be answered mathematically using the concept of expected value and information regarding the probability or the percent chance of an event occurring.

Note: The information below was created with the assistance of AI.

Level of Mathematics

This lesson is best suited for:

• High school students, particularly those in grades 9–12.

• Algebra I and Probability & Statistics classes.

• Also appropriate for middle school advanced learners (Grades 7–8) with exposure to probability concepts.

While the lesson does not involve calculus or advanced statistics, it develops key reasoning and modeling skills through applied contexts.

Application Areas

This module applies mathematics in a wide variety of real-world scenarios, including:

• Weather forecasting: Interpreting percent chance and historical probability.

• Sports analytics: Baseball batting, slugging averages, and tennis serve efficiency.

• Consumer behavior: Evaluating lotteries, raffles, and sweepstakes using expected value.

• Decision making under uncertainty: Risk assessment and fair games.

• Programming and simulation: Using basic computer programs to simulate probabilistic scenarios.

These applications make the concept of expected value relevant to both everyday decision-making and career fields (e.g., sports, finance, statistics).

Prerequisites

Students should be comfortable with:

• Basic fractions, decimals, and percentages.

• Multiplying probabilities and values to calculate expected outcomes.

• Understanding simple probability (e.g., one in N chance, complement rule).

• Reading and interpreting tree diagrams.

• Simple algebraic expressions.

• Optional: Familiarity with basic programming logic (for the included computer simulation).

Subject Matter
Core Concepts Covered

1. Expected Value

   • Introduced through relatable contexts (rainfall, batting averages).

   • Applied to sweepstakes, lotteries, and games of chance.

2. Fair and Unfair Games

   • Definition and calculation of fairness in monetary games.

   • Practical evaluation of “net gain” or “loss” scenarios.

3. Probability Modeling with Tree Diagrams

   • Complex decision trees in tennis and baseball.

   • Multi-stage outcomes visualized and calculated systematically.

4. Simulation and Approximation

   • Use of a BASIC-style computer program to compute multi-branch scenarios.

   • Early integration of computational thinking in probability.

Correlation to Mathematics Standards
Common Core State Standards (CCSS)

High School Statistics and Probability

• HSS-MD.A.1–3: Calculate expected values and use them to solve problems.

• HSS-MD.B.5: Evaluate and compare strategies using expected value.

• HSS-IC.B.6: Use simulations to informally assess probability models.

Mathematical Practices

• MP1: Make sense of problems and persevere in solving them.

• MP2: Reason abstractly and quantitatively.

• MP4: Model with mathematics.

• MP5: Use appropriate tools strategically (e.g., diagrams, programming).

• MP7: Look for and make use of structure.