Lesson 20.4

Probability and Expected Value

Introduction to the likelihood of events and the long-term average of outcomes.

Introduction

Probability quantifies uncertainty—the chance that an event occurs.Expected value tells us the average outcome over many trials. Together, they form the foundation of statistics, risk analysis, and decision-making.

Prerequisite Connection

You understand counting with permutations and combinations.

Today's Increment

We calculate probabilities and expected values for discrete events.

Why This Matters

Probability governs insurance, games, medical testing, finance, and machine learning.

Key Concepts

Probability of an Event

Always between 0 (impossible) and 1 (certain).

Complement Rule

Expected Value

Multiply each outcome by its probability, then sum.

Addition Rule (OR)

For mutually exclusive events:

Worked Examples

Example 1: Basic Probability (Basic)

A bag has 3 red, 4 blue, and 5 green marbles. Find P(red).

Total = 12 marbles, Red = 3

or 25%

Example 2: Expected Value (Intermediate)

A game costs $5. You win $20 with probability 0.2, otherwise win nothing. What's the expected value?

Outcomes: Win $15 net (20-5) with P=0.2, Lose $5 with P=0.8

(you lose $1 on average)

Example 3: Using Combinations (Advanced)

Draw 2 cards from a standard deck. P(both are aces)?

Ways to choose 2 aces:

Ways to choose any 2 cards:

Common Pitfalls

Probability > 1

If you get P > 1, something is wrong! Check your counting.

Confusing "and" with "or"

"And" usually means multiply. "Or" usually means add (minus overlap).

Ignoring cost in expected value

Always subtract the cost of playing from winnings.

Real-World Application

Insurance Pricing

Insurance companies use expected value to set premiums. If a $100,000 claim has a 0.1% chance of occurring, the expected payout is:

Premium must exceed $100 to be profitable!