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Independent Events

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Independent Events

Study Guide

Key Definition

Two events A and B are independent if $P(A \cap B) = P(A) \cdot P(B)$, meaning the occurrence of one event does not affect the probability of the other.

Important Notes

Independent events do not influence each other
If $P(A | B) = P(A)$, then A and B are independent
For independent events, $P(B | A) = P(B)$
Sample spaces can represent all possible outcomes
Understanding independence is crucial in probability

Mathematical Notation

$\cap$ represents intersection

$\cdot$ represents multiplication

$P(A)$ is the probability of event A

$P(A \cap B)$ is the probability of both A and B

$\frac{a}{b}$ represents a fraction

Remember to use proper notation when solving problems

Why It Works

Understanding that independent events do not affect each other's outcomes allows us to calculate probabilities using $P(A \cap B) = P(A) \cdot P(B)$.

Remember

For independent events, use the formula $P(A \cap B) = P(A) \cdot P(B)$.

Quick Reference

Independence:$P(A \cap B) = P(A) \cdot P(B)$

Understanding Independent Events

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Events A and B are independent if $P(A \cap B) = P(A) \cdot P(B)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

If $P(A) = \frac{1}{2}$ and $P(B) = \frac{1}{4}$, what is $P(A \cap B)$ if A and B are independent?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A student chooses a shirt and a pair of pants at random. If there are $5$ shirts and $3$ pants, what is the probability of choosing a specific shirt and a specific pair of pants?

Thinking Challenge

Thinking Exercise