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Expected Value

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Expected Value

Study Guide

Key Definition

In a probability distribution, the expected value $E(x)$ is the sum of each possible value multiplied by its probability: $E(x) = \sum x_i P(x_i)$.

Important Notes

Expected value is a type of weighted average.
It predicts the average outcome of an experiment over many trials.
The expected value may not be one of the possible outcomes.
Expected value is also known as the mean of the probability distribution.
It is useful for making predictions in probabilistic experiments.

Mathematical Notation

$\sum$ represents summation

$E(x)$ represents the expected value

$x_i$ represents a possible value of the random variable

$P(x_i)$ represents the probability of $x_i$

$=$ represents equality

Remember to use proper notation when solving problems

Why It Works

Expected value works by calculating a weighted average, which takes into account all possible outcomes and their probabilities, providing a long-term average result.

Remember

The formula $E(x) = \sum x_i P(x_i)$ is crucial for calculating the expected value.

Quick Reference

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

Understanding Expected Value

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Expected value is like an average but considers probabilities: $E(x) = \sum x_i P(x_i)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the expected value of rolling a die if you earn $1 for an odd number and $2 for an even number?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine playing a game where you win $5 with a probability of $\frac{1}{2}$ and lose $3 with a probability of $\frac{1}{2}$. What is the expected value of the game?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

In a lottery, you have a probability of $\frac{1}{1000}$ to win $1000, otherwise you lose $1. What is the expected value of playing the lottery?

Challenge Quiz

Single Choice Quiz

Advanced

A game pays $10 with a probability of $\frac{1}{4}$, $5 with a probability of $\frac{1}{2}$, and $0 with a probability of $\frac{1}{4}$. What is the expected value of this game?

Recap

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Review key concepts and takeaways