Are mutually exclusive events independent?
Short answer: If A happens, B can't, so A gives information about B. P(A∩B) = 0 ≠ P(A)P(B).
The trap
The wrong idea
“Mutually exclusive events are independent.”
It sounds reasonable, and a lot of JEE aspirants carry it into the exam. Here's a one-question check, then the explanation that makes it click.
Quick check
Try it before you peek
A and B are mutually exclusive with P(A), P(B) > 0. Are they independent?
- Yes, always
- No, never
- Sometimes
Show the answer
B. No, never
If A happens, B can't, so A gives information about B. P(A∩B) = 0 ≠ P(A)P(B).
The click
These are two different ideas. Mutually exclusive means A and B cannot happen together: P(A ∩ B) = 0. Independent means that knowing one has happened doesn't change the probability of the other: P(A ∩ B) = P(A) × P(B), or equivalently P(B | A) = P(B).
If P(A) > 0 and P(B) > 0, then P(A) × P(B) > 0, but mutual exclusivity forces P(A ∩ B) = 0. The independence condition fails. So two mutually exclusive events that both have non-zero probability are never independent; in fact they are strongly dependent.
Intuition: if A happens, you know for certain that B didn't, which is about the most informative thing A could tell you about B. Then P(B | A) = 0 while P(B) > 0. Example with one roll of a die: A = "roll a 1" and B = "roll a 2" are mutually exclusive. P(A) = P(B) = 1/6, so P(A) × P(B) = 1/36, but P(A ∩ B) = 0.
Compare independent events: roll two dice, with A = "first die shows 1" and B = "second die shows 2". These can happen together, and P(A ∩ B) = 1/36 = P(A) × P(B). Events can be both mutually exclusive and independent only if at least one of them has probability 0. Exam tip: for mutually exclusive events, P(A ∪ B) = P(A) + P(B); for independent events, P(A ∪ B) = P(A) + P(B) − P(A)P(B).
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