Data Analysis and Statistics · Term 4

Probability in Decision Making

Using probability to assess risk and make informed predictions in games and insurance.

About This Topic

Probability in Decision Making teaches Grade 7 students to apply probability concepts to real-world scenarios like games and insurance. They calculate theoretical probabilities, conduct experiments to find experimental probabilities, and use expected value to assess risk. Students explore why casinos design games with a built-in house advantage for long-term profit and how insurance companies set premiums based on predicted claim frequencies. Key questions guide them to explain fair games, where expected value equals zero, and predict long-term outcomes aligning with theoretical probabilities.

This topic builds on data analysis and statistics by introducing the law of large numbers, showing how repeated trials make experimental results approach theoretical values. It fosters skills in prediction, risk evaluation, and informed choices, connecting math to financial literacy and consumer decisions in Ontario contexts like lotteries or auto insurance.

Active learning benefits this topic greatly because simulations of games or insurance claims allow students to run hundreds of trials quickly, witnessing probability patterns emerge firsthand. Group challenges in designing fair games reinforce calculations through peer review and testing, making abstract ideas concrete and boosting retention.

Learning Objectives

• Calculate the theoretical probability of simple and compound events to predict long-term frequencies.
• Compare theoretical and experimental probabilities from simulations to evaluate the fairness of games.
• Analyze how casinos and insurance companies use expected value to ensure profitability and manage risk.
• Design a simple game and explain the probability-based strategies used to make it fair or to create a house advantage.

Before You Start

Introduction to Probability

Why: Students need to understand basic probability concepts, including calculating the probability of single events and identifying possible outcomes.

Data Collection and Representation

Why: Students must be able to collect data from experiments and represent it, often in tables or graphs, to compare theoretical and experimental probabilities.

Key Vocabulary

Theoretical Probability	The likelihood of an event occurring based on mathematical calculations, assuming all outcomes are equally likely.
Experimental Probability	The likelihood of an event occurring based on the results of an experiment or simulation, calculated as the ratio of favorable outcomes to total trials.
Expected Value	The average outcome of an event if it were repeated many times, calculated by multiplying each possible outcome by its probability and summing the results.
Fair Game	A game where the expected value for each player is zero, meaning over many plays, no player is expected to win or lose money.
Law of Large Numbers	A principle stating that as the number of trials in a probability experiment increases, the experimental probability will approach the theoretical probability.

Active Learning Ideas

Pairs Activity: Design Fair and Unfair Games

Pairs use dice or cards to invent two games, one fair and one with house edge. They calculate probabilities and expected values, then swap games with another pair for 50 trials and compare results to predictions. Discuss adjustments for fairness.

⏱ 45 min·Pairs

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Small Groups: Insurance Premium Simulation

Groups role-play as an insurance company over 20 'policyholders' using random draws for claims. They calculate average claims, set premiums to ensure profit, and run multiple rounds to see long-term stability. Graph results to analyze risk.

⏱ 50 min·Small Groups

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Whole Class: Casino Spinner Trials

Create class spinners with unequal sections representing casino payouts. Predict long-term frequencies, then conduct 100 collective spins using a randomizer app or physical spinner. Tally results on a shared chart and compute house edge.

⏱ 40 min·Whole Class

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Individual: Long-Term Prediction Journal

Students flip coins or roll dice 100 times individually, recording outcomes daily over a week. Calculate running experimental probabilities and compare to theoretical 0.5. Reflect on law of large numbers in a journal entry.

⏱ 30 min·Individual

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Real-World Connections

Casinos like Fallsview Casino Resort in Niagara Falls use probability to set the odds for games like blackjack and roulette, ensuring a consistent profit margin through the house advantage.

Insurance companies, such as State Farm or Aviva Canada, use actuarial science, a field heavily reliant on probability, to calculate premiums for auto or home insurance based on the likelihood of claims.

Lottery corporations, like the Ontario Lottery and Gaming Corporation (OLG), design games with extremely low probabilities of winning large prizes, making them profitable while offering a small chance of significant reward to players.

Watch Out for These Misconceptions

Common Misconception

A single outcome or short streak determines true probability.

What to Teach Instead

The law of large numbers shows experimental probabilities approach theoretical values over many trials. Group simulations with 100+ trials help students plot data and observe convergence, correcting reliance on small samples through visual evidence.

Common Misconception

Fair games always have a 50% win chance for each player.

What to Teach Instead

Fairness means zero expected value, possible with unequal probabilities if payouts balance. Pairs testing varied games calculate expected values and adjust payouts, revealing this through trial data and peer explanations.

Common Misconception

Casinos and insurers profit by cheating or fixing results.

What to Teach Instead

Profit comes from mathematical house edges and premium calculations based on probability. Whole-class spinner challenges demonstrate consistent long-term gains without manipulation, as students verify through repeated, transparent trials.

Assessment Ideas

Quick Check

Present students with a scenario: 'A spinner has 4 equal sections: red, blue, green, yellow. What is the theoretical probability of landing on red? If you spin it 20 times and land on red 7 times, what is the experimental probability?' Ask students to write their answers and show their calculations.

Discussion Prompt

Pose the question: 'Imagine a simple dice game where Player A wins if they roll a 6, and Player B wins if they roll any other number. Is this game fair? Explain your reasoning using the concept of expected value and theoretical probability.'

Exit Ticket

Ask students to write one sentence explaining how a casino uses probability to make money and one sentence explaining how an insurance company uses probability to set prices.

Frequently Asked Questions

How do casinos use probability to ensure profitability?

Casinos design games with a house edge, where payouts are less than true odds, ensuring positive expected value over many plays. For example, in roulette, the zero slot gives the house about 5% advantage. Students simulate spins to see long-term wins accumulate, even after short-term losses, mirroring real casino math.

What does it mean for a game to be mathematically fair?

A fair game has an expected value of zero for players, meaning long-term average gains and losses balance. Students calculate this by multiplying outcomes by probabilities and summing. Testing custom games shows fair ones neither favor house nor player consistently.

How can active learning help students grasp probability in decision making?

Active simulations like group insurance claims or spinner trials let students run hundreds of events, observing law of large numbers in action. Collaborative game design requires applying expected value formulas amid peer feedback. These hands-on methods make risk assessment tangible, improve prediction accuracy, and connect theory to decisions like buying insurance.

How do insurance companies use probability for predictions?

Insurers analyze historical data to find claim probabilities, set premiums covering average claims plus profit margin. For car insurance, they predict accident frequencies by age or region. Classroom simulations with random claims teach students how large sample sizes refine these predictions for sustainability.