Direct comparison
Independent vs Mutually Exclusive Events
Independent vs mutually exclusive: the defining formulas, how to test each, and worked textbook examples showing why the two ideas are not the same.
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How do Independent events, Mutually exclusive events compare side by side?
The table below compares Independent events, Mutually exclusive events across 11 procurement-relevant dimensions, from plain-language meaning through source.
Side-by-side comparison
| Dimension | Independent events | Mutually exclusive events |
|---|---|---|
| Plain-language meaning | Knowing that one event occurred does not affect the chance that the other occurs. | The two events cannot occur at the same time. |
| What it says about outcomes | Nothing about whether the events share outcomes. Independent events can overlap. | A and B share no outcomes. |
| Defining condition | P(A AND B) = P(A) × P(B). | P(A AND B) = 0. |
| How to test it (any one check suffices for independence) | Check P(A|B) = P(A), or P(B|A) = P(B), or P(A AND B) = P(A) × P(B). | Check whether any outcome belongs to both events, or whether P(A AND B) = 0. |
| Effect on the multiplication rule, P(A AND B) = P(B) × P(A|B) | P(A|B) = P(A), so the rule reduces to P(A AND B) = P(A) × P(B). | The product is 0, because P(A AND B) = 0. |
| Effect on the addition rule, P(A OR B) = P(A) + P(B) − P(A AND B) | The overlap term is subtracted and equals P(A) × P(B), so P(A OR B) = P(A) + P(B) − P(A) × P(B). | The overlap term is 0, so P(A OR B) = P(A) + P(B). |
| Can the other property hold at the same time? | If both events have positive probability, P(A) × P(B) is greater than 0, so they cannot be mutually exclusive. | If both events have positive probability, P(A AND B) = 0 is not equal to P(A) × P(B), so they are dependent. (Derived from the two formulas.) |
| Worked example from the textbook | P(G) = 0.6, P(H) = 0.5, P(G AND H) = 0.3. Since 0.6 × 0.5 = 0.3, G and H are independent. | A = heads then an even number, P(A) = 3/12; B = heads then a three, P(B) = 1/12. P(A AND B) = 0, so they are mutually exclusive, and 0 is not (3/12)(1/12), so they are also dependent. |
| Counter-example from the textbook | P(W) = 0.60, P(L) = 0.50, P(W AND L) = 0.45. Since 0.45 is not 0.30, W and L are not independent. | In a box of cards, P(R) = 3/8 and P(B) = 5/8 with P(R AND B) = 0: R and B are mutually exclusive. |
| When to use which | Use the independence test when the question is whether learning one event changes the probability of the other. | Use the mutually exclusive property when the question is whether the two events can both happen in one trial. |
| Source | OpenStax, Introductory Statistics 2e, section 3.2 (Illowsky and Dean), read 2026-10-29; OpenStax, Introductory Statistics 2e, section 3.3 (Illowsky and Dean), read 2026-10-29. | OpenStax, Introductory Statistics 2e, section 3.2 (Illowsky and Dean), read 2026-10-29; OpenStax, Introductory Statistics 2e, section 3.3 (Illowsky and Dean), read 2026-10-29. |
Common questions
Common questions about Independent events vs Mutually exclusive events
Are independent and mutually exclusive the same thing?
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No. OpenStax states that independent and mutually exclusive do not mean the same thing. Independence is about whether one event changes the probability of the other; mutual exclusivity is about whether the events can occur together.
Can two events be both independent and mutually exclusive?
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Only if at least one of them has probability 0. Mutual exclusivity gives P(A AND B) = 0, and independence requires P(A AND B) = P(A) × P(B), so both can hold only when P(A) × P(B) = 0. This follows from the two formulas; it is not a sentence quoted from the textbook.
Does P(A AND B) = 0 mean the events are independent?
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No. P(A AND B) = 0 means the events are mutually exclusive. Independence needs P(A AND B) to equal P(A) × P(B), which is greater than 0 when both events are possible.
Going deeper
