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Probability

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Probability

Study Guide

Key Definition

Probability is a measure of the likelihood of an event occurring, represented by a number between $0$ and $1$.

Important Notes

The probability of an impossible event is $0$.
The probability of a certain event is $1$.
The sum of probabilities of all possible outcomes is $1$.
Experimental probability approaches theoretical probability with more trials.
Use the formula $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$.

Mathematical Notation

$P(A)$ probability of event A

$×$ multiplication (used for independent events)

$∪$ union (or) of two events

$∩$ intersection (and) of two events

Remember to use proper notation when solving problems

Why It Works

When all outcomes are equally likely, each has probability $1/N$, so an event with k favorable outcomes has $P(\text{event})=\frac{k}{N}$. For two events A and B, $P(A∪B)=P(A)+P(B)−P(A∩B)$ avoids double counting, and if independent, $P(A∩B)=P(A)×P(B)$. Compound events often use these rules. Conditional probability is defined by $P(A\mid B)=\frac{P(A∩B)}{P(B)}$, forming the basis for Bayes’ theorem.

Remember

Probability ranges from $0$ to $1$, where $0$ is impossible and $1$ is certain.

Quick Reference

Probability Formula:$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$

Understanding Probability

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Probability measures how likely an event is to occur. For equally likely outcomes, $P(\text{event})=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$. For example, drawing a red ball from a bag with 2 red and 3 blue balls: $P(\text{red})=\frac{2}{5}=0.4$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the probability of choosing the letter $K$ from the set {$J, K, L, M, N$}?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

In a game, the probability of winning a prize is $\frac{1}{10}$. If you play $20$ times, how many prizes do you expect to win?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a bag contains $3$ red balls and $7$ blue balls, what is the probability of drawing a red ball?

Challenge Quiz

Single Choice Quiz

Advanced

What is the probability of rolling an odd number on a fair six-sided die that isn't a $3$?

Recap

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Review key concepts and takeaways