Introduction to Probability

What is Probability?

Probability is a measure of how likely an event is to occur. It’s expressed as a number between 0 and 1 (or 0% and 100%):

• P = 0 → Impossible event
• P = 1 → Certain event
• 0 < P < 1 → Uncertain event

Basic Probability Terms

Before diving into calculations, let’s define key terms:

Term	Definition	Example (Rolling a Die)
Experiment	A process with uncertain outcomes	Rolling a die
Outcome	A single result of an experiment	Rolling a 4
Sample Space (S)	All possible outcomes	{1, 2, 3, 4, 5, 6}
Event	A subset of outcomes	Rolling an even number {2, 4, 6}

Calculating Basic Probability

For equally likely outcomes, probability is calculated as:

Probability Rules

Rule 1: Complement Rule

The probability of an event NOT occurring is:

Rule 2: Addition Rule

For the probability of A OR B occurring:

Rule 3: Multiplication Rule

For the probability of A AND B occurring:

Rule 4: Conditional Probability

The probability of A occurring GIVEN that B has occurred:

Types of Events

Independent Events

Events where the occurrence of one doesn’t affect the probability of the other.

Test: P(A ∩ B) = P(A) × P(B)

Examples:

• Flipping a coin twice
• Rolling two dice
• Drawing cards WITH replacement

Dependent Events

Events where the occurrence of one affects the probability of the other.

Test: P(A ∩ B) ≠ P(A) × P(B)

Examples:

• Drawing cards WITHOUT replacement
• Selecting people from a group without returning them

Mutually Exclusive Events

Events that cannot occur at the same time.

Test: P(A ∩ B) = 0

Examples:

• Rolling a 1 AND a 6 on one die
• Being exactly 20 AND exactly 30 years old

Common Probability Mistakes

Probability Summary Table

Concept	Formula	Use When
Basic probability	P(A) = n(A)/n(S)	Counting equally likely outcomes
Complement	P(A’) = 1 - P(A)	Finding “not A”
Addition (exclusive)	P(A∪B) = P(A) + P(B)	Events can’t overlap
Addition (general)	P(A∪B) = P(A) + P(B) - P(A∩B)	Events may overlap
Multiplication (independent)	P(A∩B) = P(A) × P(B)	Events don’t affect each other
Conditional	P(A|B) = P(A∩B) / P(B)	One event given another occurred

Practice Problem

A bag contains 5 red balls and 3 blue balls. You draw 2 balls without replacement.

Question: What’s the probability of drawing 2 red balls?

Click for solution

• P(1st red) = 5/8
• P(2nd red | 1st red) = 4/7

P(both red) = (5/8) × (4/7) = 20/56 = 5/14 ≈ 0.357 or 35.7%

Summary

In this lesson, you learned:

• Probability measures likelihood on a scale from 0 to 1
• The complement rule: P(A’) = 1 - P(A)
• The addition rule for “or” probabilities
• The multiplication rule for “and” probabilities
• The difference between independent and dependent events
• How to calculate conditional probability

Try It Yourself

Use our Probability Calculator to practice different probability calculations!

Next Steps

• Probability Distributions - Discrete and continuous distributions
• Bayes’ Theorem - Advanced conditional probability