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Using Probabilities to Make Fair Decisions

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Using Probabilities to Make Fair Decisions

Study Guide

Key Definition

A probability experiment is considered "fair" if all outcomes are equally likely.

Important Notes

A fair decision means each outcome has an equal chance.
In a fair game, the expected value is $0$.
Use a number cube for equal probability among six outcomes.
Flipping a fair coin yields equal probabilities for heads and tails; however, using coin flips to choose among an odd number of options may require additional steps to maintain fairness.
Check each method for fairness by analyzing probabilities.

Mathematical Notation

$\frac{1}{6}$ represents the probability of one out of six outcomes

$+$ represents addition

$E[X]$ denotes the expected value of random variable X

$\Sigma$ denotes summation

$P(A)$ denotes the probability of event A

Remember to use proper notation when solving problems

Why It Works

Fair decisions rely on equal probability distribution, which ensures that each participant has an equal chance, often evaluated using $\frac{1}{n}$ where $n$ is the number of outcomes.

Remember

Fairness implies equal probability for each outcome: $P(outcome) = 1/n$.

Quick Reference

Probability of One Out of Six:$\frac{1}{6}$

Expected Value:$E[X] = \Sigma xP(X = x)$

Fairness Criterion:$P(i) = 1/n$ for each outcome i

Understanding Using Probabilities to Make Fair Decisions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A fair decision gives each option the same chance. For n options, each has probability $1/n$. For example, a six-sided die gives each face a $1/6$ chance.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which method ensures a fair decision for choosing a captain among 6 players?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are organizing a game night and need to choose a game leader fairly among 8 friends. Question: Propose a method and calculate the probability for each friend.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a spinner with three colors — red (probability 0.5), blue (0.3), and green (0.2). How would you adjust the system to make it fair among the three colors?

Challenge Quiz

Single Choice Quiz

Advanced

In a game with multiple outcomes, how can you calculate the expected value to determine fairness?

Recap

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Review key concepts and takeaways