What is the difference between independent events and mutually exclusive events?
Mutually exclusive events cannot both occur at once — knowing one happened tells you the other didn't. Independent events can both occur, but knowing one happened gives no information about the other. These concepts are nearly opposite: non-trivial mutually exclusive events are always dependent.
How to think about it
This distinction trips up more candidates than almost any other basic probability question. The key is understanding what each condition says about information flow between events.
Mutually exclusive (disjoint)
Events A and B are mutually exclusive when they share no outcomes:
P(A ∩ B) = 0Consequence for union: P(A ∪ B) = P(A) + P(B) (no double-counting needed).
Example: rolling a single die — getting a 2 and getting a 5 on the same roll are mutually exclusive.
Independent
Events A and B are independent when knowing B occurred gives no update to A:
P(A | B) = P(A) ⟺ P(A ∩ B) = P(A) × P(B)Example: flipping two separate coins — the result of the first flip tells you nothing about the second.
Why they are nearly opposite
If A and B are mutually exclusive and P(A) > 0 and P(B) > 0, then:
P(A | B) = P(A ∩ B) / P(B) = 0 / P(B) = 0 ≠ P(A)So knowing B occurred eliminates A — the strongest possible dependence. The only edge case: if either event has probability zero, they can be both mutually exclusive and independent in a trivial mathematical sense, but that is not meaningful in practice.
Quick comparison
| Property | Mutually exclusive | Independent |
|---|---|---|
P(A ∩ B) | 0 | P(A)×P(B) |
P(A | B) | 0 | P(A) |
| Can both happen? | No | Yes |
| Are they dependent? | Yes (if non-trivial) | No |