Mathematics · Ch 13 — Probability

Independent Events

13.4

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:

P(E) = \frac{13}{52} = \frac{1}{4}, \quad P(F) = \frac{4}{52} = \frac{1}{13}

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

P(E|F) = \frac{P(E \cap F)}{P(F)} = \frac{1/52}{1/13} = \frac{1}{4} = P(E)

P(F|E) = \frac{P(E \cap F)}{P(E)} = \frac{1/52}{1/4} = \frac{1}{13} = P(F)

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

Note

Independence is about probability, not about the events being causally unrelated. Two events can be independent in probability even if they are logically connected, as long as the numbers work out.

Formal Definition of Independence

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

P(F|E) = P(F) \;\; (P(E) \neq 0) \quad \text{and} \quad P(E|F) = P(E) \;\; (P(F) \neq 0)

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:

P(E \cap F) = P(E) \cdot P(F)

This gives the second, more commonly used definition.

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

P(E \cap F) = P(E) \cdot P(F)

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

Watch out

Do not confuse independent events with mutually exclusive events. If $E$ and $F$ are mutually exclusive with non-zero probabilities, then $P(E \cap F) = 0$ but $P(E) \cdot P(F) > 0$ , so they cannot be independent — and vice versa.

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

Property	Independent Events	Mutually Exclusive Events
Defined in terms of	Probability of events	Events as subsets of sample space
Common outcomes	May have common outcomes	Never have common outcomes
Non-zero probability case	$P(E \cap F) = P(E)P(F) > 0$	$P(E \cap F) = 0$

(iii) Independence of Experiments

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

$P(E \cap F) = P(E) \cdot P(F)$ …

Definition 2Independent Events

Definition of Independent Events

Two events $E$ and $F$ from the same random experiment are independent if the occurrence of one does not affect the probability of the other.

Formal definition (NCERT Definition 3):

$E$ and $F$ are independent if and only if

P(E \cap F) = P(E) \cdot P(F)

Equivalent definition (when probabilities are non-zero):

If $P(E) \neq 0$ and $P(F) \neq 0$ , then $E$ and $F$ are independent if

P(E|F) = P(E) \quad \text{and} \quad P(F|E) = P(F)

Important conditions and remarks:

If $P(E \cap F) \neq P(E) \cdot P(F)$ , the events are dependent.
Independent ≠ Mutually exclusive.
- Mutually exclusive events have no common outcome ( $E \cap F = \emptyset$ ).
- Independent events can have common outcomes.
- Two independent events with non-zero probabilities cannot be mutually exclusive, and vice versa.
For three events $A, B, C$ to be mutually independent, all four conditions must hold:

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

P(A \cap C) = P(A)P(C)

P(B \cap C) = P(B)P(C)

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

Intuition

Independence means "knowing that one event happened gives you no new information about whether the other event will happen."

Tiny Concrete Example …

Definition 3Independent Events