Probability Calculator · Practical guide
Mutually Exclusive vs Independent Events
Exclusive events cannot overlap; independent events overlap exactly as chance predicts. The formulas differ.
At a glance
The One-Line Difference
Mutually exclusive events cannot both occur — P(A AND B) = 0. Independent events do not influence each other — P(A AND B) = P(A) × P(B). One describes overlap, the other describes influence, and they are not the same thing.
Interactive tool
Probability Calculator
Enter your own values, then compare the result with the examples below.
Same Numbers, Different Answers
Take P(A) = 0.3 and P(B) = 0.4 and apply each relationship:
| Quantity | Mutually Exclusive | Independent |
|---|---|---|
| P(A AND B) | 0 | 0.3 × 0.4 = 0.12 |
| P(A OR B) | 0.3 + 0.4 = 0.70 | 0.3 + 0.4 − 0.12 = 0.58 |
The calculator computes both rows — just switch the relationship selector between Mutually Exclusive and Independent.
Why Exclusive Events Are Not Independent
If two events cannot happen together, then learning one occurred drops the other's probability to zero — which means they do affect each other. Rolling a 1 and rolling a 6 on the same die are mutually exclusive but strongly dependent. The only way to be both exclusive and independent is for one event to have probability 0.
Real-World Examples
- Mutually exclusive: rolling a 1 versus an even number on one die roll; drawing a heart versus a spade from one card.
- Independent: two coin flips; a die roll and a coin flip; drawing two cards with replacement.
- Neither: drawing two cards without replacement — the first draw changes the second's odds.
Frequently Asked Questions
Can two events be both mutually exclusive and independent?
Only in the trivial case where one has probability 0. Otherwise, exclusivity means one event occurring wipes out the other — a clear dependence.
What is P(A or B) for mutually exclusive events?
Just the sum, P(A) + P(B). With no overlap to subtract, the addition rule simplifies to pure addition.
Is rolling a 1 and rolling an even number mutually exclusive?
Yes, on a single roll — one die cannot show both at once. Across two separate rolls they would be independent instead.
How do I know which relationship applies to my problem?
Ask two questions: can both happen at once, and does one outcome change the other's odds? No to the first means exclusive; no to the second means independent; a yes to either points to the general conditional form.