Independent Events

13.4 Independent Events

The Core Idea: When One Event Doesn't Affect Another

Draw a single card from a well-shuffled deck of 52. Let $E$ be "the card is a spade" and $F$ be "the card is an ace." With 13 spades and 4 aces:

The event $E \cap F$ is "the ace of spades," so $P(E \cap F) = \frac{1}{52}$. Then:

The occurrence of one event has not changed the probability of the other. Events with this property are called independent events.

Formal Definition of Independence

Conditional form: Two events $E$ and $F$ are independent if

The Multiplication Rule for Independent Events

From the multiplication rule $P(E \cap F) = P(E) \cdot P(F|E)$, independence gives $P(F|E) = P(F)$, so:

This gives the second, more commonly used definition.

Product form: Two events $E$ and $F$ associated with the same experiment are independent if

Unlike the conditional form, this does not require $P(E) \neq 0$ or $P(F) \neq 0$ — it works in all cases.

Important Remarks on Independence

(i) Dependent Events

$E$ and $F$ are dependent if they are not independent, i.e. $P(E \cap F) \neq P(E) \cdot P(F)$.

(ii) Independence vs. Mutual Exclusivity

(iii) Independence of Experiments

Two experiments are independent if for every pair of events $E$ (from the first) and $F$ (from the second),

$P(E \cap F) = P(E) \cdot P(F)$ …