Independent vs Mutually Exclusive Events

Independent vs Mutually Exclusive Events — Moosa Academy

Most probability mistakes come from choosing the wrong operation. The trick is a single question: are there several events (multiply), or one event with alternative outcomes (add)? "And" means multiply; "or" means add.

Golden rule One question decides it

Ask: is there more than one event?

Yes — "and": more than one event (independent or dependent) → multiply → the probability goes down.

No — "or": one event with alternative outcomes (mutually exclusive or not) → add → the probability goes up.

Formulas The four laws

More than one event — multiply

Independent:

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

Dependent:

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

One event, alternatives — add

Mutually exclusive (never together, so $P(A \cap B) = 0$ ): $P(A \cup B) = P(A) + P(B)$ .

Not mutually exclusive: subtract the overlap, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ .

Example A box of 3 balls

Multiply — first green AND second yellow

Two separate draws (with replacement) — the word is "and".

\tfrac{1}{3} \times \tfrac{1}{3} = \tfrac{1}{9}

⟹ ≈ 11.1% — smaller, because we multiplied.

Add — one draw, green OR yellow

One draw, two acceptable outcomes — the word is "or", and they cannot both happen.

\tfrac{1}{3} + \tfrac{1}{3} = \tfrac{2}{3}

⟹ ≈ 66.7% — larger, because we added.

Add to a certainty — 5 red and 3 blue

A single ball is red or blue — the only two options.

\tfrac{5}{8} + \tfrac{3}{8} = 1

⟹ P = 1, a certain event.

Reference The full comparison

Feature	Independent / Dependent	Mutually exclusive / Not
Number of events	more than one	one, with alternative outcomes
Keyword	"and"	"or"
Operation	× multiply	+ add
Effect	probability decreases	probability increases
Example (3 balls)	1/3 × 1/3 = 1/9	1/3 + 1/3 = 2/3

Summary

Key question: "more than one event?" — yes → multiply, no → add.
Independent: $P(A \cap B) = P(A) \times P(B)$ ; dependent: $P(A \cap B) = P(A) \times P(B \mid A)$ .
Mutually exclusive: $P(A \cup B) = P(A) + P(B)$ ; not mutually exclusive: subtract the overlap $P(A \cap B)$ .
Multiplying makes the probability smaller; adding makes it larger.