Independence vs disjointness

These two concepts confuse even experienced students. They sound related. They're almost opposites.

What do you think?

'The events don't affect each other' vs 'The events can't happen together': which describes independence?

Disjoint events

Disjoint Events

Events A and B are disjoint (or mutually exclusive) if they cannot both occur: $A \cap B = \emptyset$ which means $P(A \cap B) = 0$ .

Examples of disjoint events:

Rolling a 1 and rolling a 6 (same die, same roll)
Being born in January and being born in July
A coin landing heads and landing tails (same flip)

True or False: 'Drawing a King' and 'Drawing a Queen' from the same card are disjoint. (true or false)

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Independent events

Independent Events

Events A and B are independent if knowing one happened doesn't change the probability of the other: $P(A|B) = P(A)$ Equivalently: $P(A \cap B) = P(A) \cdot P(B)$

Examples of independent events:

Flipping heads on one coin and heads on another
It raining in Tokyo and you rolling a 6 in New York
Drawing a card, replacing it, and drawing another

P(A) = 0.3, P(B) = 0.5. If A and B are independent, what's P(A ∩ B)? (decimal, e.g. 0.42)

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The key difference

What do you think?

If A and B are disjoint (can't both happen), are they independent?

Disjoint events are maximally DEPENDENT, not independent!

If A and B are disjoint and B happens, then A definitely didn't happen. That's not independence. That's the strongest possible dependence!

Proof: Disjoint ≠ Independent

P(A|B) = \frac{P(A \cap B)}{P(B)}

Start with the definition of conditional probability.

Step 1 of 3

Visual intuition

Drag the circles to see the difference:

Independence vs Disjointness

Sample Space

A ∩ B

Dependent

The events influence each other

Disjoint: Circles don't touch. If you're in B, you're definitely not in A.
Independent: Circles can overlap. The fraction of B that overlaps A equals P(A).

The independence test

To check if events are independent, verify:

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

Try checking independence yourself with different card events:

Independence Checker

Event A

Event B

P(A)26/52 = 1/2

P(B)4/52 = 1/13

P(A) × P(B)0.0385

P(A ∩ B)0.0000

Not Independent

P(A ∩ B) ≠ P(A) × P(B)

Independence Sampler

P(A) = 0.3, P(B) = 0.4 — sample 100 events at a time

What do you think?

A = 'red card' (P = 1/2), B = 'ace' (P = 1/13). Are they independent?

A = 'face card' (P = 12/52), B = 'spade' (P = 13/52). P(A ∩ B) = 3/52. Is P(A)P(B) = 3/52? (yes/no) (yes or no)

Conditional independence

Events can be:

Independent overall, but dependent given some condition
Dependent overall, but independent given some condition

Let:

A = "smoke detector goes off"
B = "sprinklers activate"

These events are dependent (both happen when there's fire). But conditional on there being a fire, they might be independent since each system responds to the fire separately.

Conditional Independence

Smoke detector (A) and sprinklers (B) — are they independent?

P(A∩B)

0.0100

P(A) · P(B)

0.0017

≈ Equal — conditionally independent (both triggered by hidden cause: fire)

Why it matters

Confusing independence and disjointness leads to errors:

What do you think?

P(A) = 0.3, P(B) = 0.4, and A & B are DISJOINT. What's P(A ∪ B)?

P(A) = 0.3, P(B) = 0.4, INDEPENDENT. P(A ∪ B) = ? (decimal, e.g. 0.42)

P(A) = 0.3, P(B) = 0.4, DISJOINT. P(A ∪ B) = ? (decimal, e.g. 0.42)

Summary

Pick the Formula

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P(A)=0.1, P(B)=0.9, independent

Property	Disjoint	Independent
Definition	Can't both happen	Don't influence each other
$P(A \cap B)$	$= 0$	$= P(A) \cdot P(B)$
$P(A	B)$	$= 0$
Venn diagram	No overlap	Proportional overlap
Dependence	Maximally dependent	Zero dependence

Quick test: "If I learn B happened, does that change what I think about A?"

Yes → Dependent (includes disjoint as extreme case)
No → Independent

Test your understanding

True or False: If A and B are disjoint with P(A) > 0 and P(B) > 0, they cannot be independent. (true or false)

A die roll: A = 'even', B = 'greater than 4'. P(A) = 1/2, P(B) = 1/3, P(A ∩ B) = 1/6. Independent? (yes/no) (yes or no)

Same die: A = 'even', C = 'equal to 3'. Are A and C disjoint? (yes/no) (yes or no)

P(A) = 0.2, P(B) = 0.5, events are INDEPENDENT. What's P(A ∩ B)? (decimal, e.g. 0.42)

P(A) = 0.2, P(B) = 0.5, events are DISJOINT. What's P(A ∩ B)? (whole number)

What's next

You've seen that independence and disjointness are different. But there's another twist: events can be independent overall yet become dependent when you condition on a third event. That's conditional independence, and it's central to Bayesian networks, Markov chains, and every spam filter you've ever used.