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Using Expected Values to Make Decisions

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Using Expected Values to Make Decisions

Study Guide

Key Definition

The expected value is the average amount one can expect to win or lose per trial in a probability experiment. It is calculated using the formula $E(X) = \sum (x_i \times P(x_i))$.

Important Notes

Expected value helps in making informed decisions.
A positive expected value indicates a likely gain.
A negative expected value suggests a probable loss.
Equal probabilities mean that each outcome is equally likely.
Always consider the sample space when calculating probabilities.

Mathematical Notation

$\sum$ represents summation

$\times$ represents multiplication

$\frac{a}{b}$ represents a division of a by b

$E(X)$ represents the expected value

Remember to use proper notation when solving problems

Why It Works

Expected values provide a long-term average outcome of random events, guiding decisions by quantifying potential risks and rewards with probability.

Remember

Expected value is a key concept in probability, informing decisions by averaging possible outcomes via $E(X) = \sum x_i \times P(x_i)$.

Quick Reference

Expected Value Formula:$E(X) = \sum x_i \times P(x_i)$

Understanding Using Expected Values to Make Decisions

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation with payoffs incorporated: $E(X) = 2 \times (1/6) + 5 \times (1/6) + (-1) \times (4/6)$

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the expected value if you win $2$ tokens with probability $\frac{1}{6}$ and win $0$ tokens otherwise?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Consider a game where a player wins $5$ tokens with probability $\frac{1}{6}$, or loses $1$ token (the entry fee) otherwise. Should you play?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a spinner with six equally likely sectors: red, blue, and four others. Red wins $2$ tokens, blue wins $5$ tokens, others lose $1$ token each. Should you play?

Challenge Quiz

Single Choice Quiz

Advanced

If a game costs 1 token to play, and you have an expected value of $0.5$ tokens per play, what is your average gain per play?

Recap

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Review key concepts and takeaways