Independence of Events — Mathematics Knowledge System

Overview

Two events A and B are independent if the occurrence of one does not change the probability of the other: P(A|B) = P(A), or equivalently P(A ∩ B) = P(A)P(B). This multiplicative rule is the defining test for independence, and it extends to collections of more than two events, where mutual independence requires the product rule to hold for every subcollection, not just pairwise. Independence is a modeling assumption, not something you can verify from a single outcome — it says that learning about B gives no information about A. Independent events are distinct from mutually exclusive events; in fact, two events with positive probability cannot be both mutually exclusive and independent.

Intuition

Independence means the two events run on separate 'information tracks.' Flipping a coin and rolling a die are independent because finding out the coin landed heads tells you absolutely nothing about what the die will show. Contrast this with drawing two cards from a deck without replacement: knowing the first card was an ace changes the odds for the second draw, so those events are dependent. The multiplication rule P(A∩B) = P(A)P(B) is just the arithmetic signature of 'no information leaks between the two events.'

Formal Definition

Definition

Events A and B are independent if and only if:

P(A \cap B) = P(A)\,P(B)

Independence (product rule)

P(A \mid B) = P(A) \quad \text{(when } P(B) > 0\text{)}

Equivalent conditional form

P(A_1 \cap A_2 \cap \cdots \cap A_n) = \prod_{i=1}^n P(A_i)

Mutual independence of n events

Worked Examples

The coin flip and die roll are independent, so multiply their individual probabilities.
$P(\text{H}) = \tfrac{1}{2}, \quad P(4) = \tfrac{1}{6}$
Apply the product rule for independent events.
$P(\text{H} \cap 4) = \tfrac{1}{2}\cdot\tfrac{1}{6} = \tfrac{1}{12}$

Answer: P(heads and a 4) = 1/12.

Practice Problems

Difficulty 3/10

Two fair coins are flipped independently. Find the probability both land heads.

Difficulty 4/10

In a group, P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. Are A and B independent?

Difficulty 5/10

A system has two independent backup components, each with a 0.9 probability of working. What is the probability that BOTH fail (so the whole backup system fails)?

Quiz

Summary

A and B are independent exactly when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A).
Independence means no information flows between the events; it is distinct from mutual exclusivity.
Mutual independence of n events requires the product rule to hold for every subcollection, not just pairs.

References

WebsiteWikipedia — Independence (probability theory)