Independent vs Mutually Exclusive Events: What Is the Difference?

The two definitions side by side

Two events A and B can be related in two different ways. They can affect each other's probability (independence), or they can rule each other out (mutual exclusivity). The first is about how likely one event is once the other is known; the second is about whether both can happen at all.

The same probabilities in both situations

Take two events that each have probability 1/6. In the first situation they are a six on the first roll of a die and a six on the second roll, which are independent. In the second they are a six and a five on one roll, which are mutually exclusive. The probabilities of the single events are the same; everything else differs.

With independent events P(A or B) = 1/6 + 1/6 − 1/36 = 11/36, because both can happen and the overlap is counted once. With mutually exclusive events there is no overlap to subtract, so P(A or B) = 1/6 + 1/6 = 1/3. Enter 1/6 and 1/6 in the probability of multiple events calculator and switch between "Independent" and "Mutually exclusive" to see both columns.

Why mutually exclusive events are not independent

It is tempting to think that events which do not affect each other must be mutually exclusive, or the other way round. In fact the two properties pull in opposite directions.

• Mutually exclusive events with positive probabilities are dependent. If a die shows a five, it cannot show a six. So P(six given five) = 0, while P(six) = 1/6. Learning that B happened changed the probability of A, which is exactly what dependence means.
• Independent events with positive probabilities can happen together. If P(A) > 0 and P(B) > 0, independence gives P(A and B) = P(A) × P(B) > 0, so the two events overlap and are not mutually exclusive.

The two properties can only meet when a probability is 0. Mutually exclusive means P(A and B) = 0 and independent means P(A and B) = P(A) × P(B), so both hold only if P(A) × P(B) = 0. An event that never happens, such as rolling a 7 on a standard die, is independent of every event and mutually exclusive with every event. That is a technical exception and not a useful case.

How to tell which one you have

1. Can both events happen in the same trial? If not, they are mutually exclusive, and therefore dependent whenever both have a chance of happening.
2. If they can overlap, find P(A and B). Count the outcomes in both events, or use the probability you are given.
3. Compare it with P(A) × P(B). Equal means independent; different means dependent. A larger P(A and B) means the events tend to happen together, a smaller one that they tend to avoid each other.

The table applies the steps to four situations.

For a king and a heart, P(A or B) = 1/13 + 1/4 − 1/52 = 4/13 ≈ 0.307692. For a king or a queen there is nothing to subtract: 1/13 + 1/13 = 2/13 ≈ 0.153846. For the third row P(A or B) = 0.4 + 0.5 − 0.3 = 0.6, and P(A given B) = 0.3 / 0.5 = 0.6, which differs from P(A) = 0.4, so knowing B raises the probability of A.

Formulas for both cases

The general rules always hold; the shortcuts apply only when you know the relationship. When you know P(A), P(B) and P(A and B), the general formulas need no assumption at all. For the probability of A when B is known, see the conditional probability calculator, and for how a probability is updated by evidence, Bayes' theorem explained.

Dependent events that are not exclusive

Most real events are neither independent nor mutually exclusive: they can happen together, but one changes the chance of the other. Drawing cards without replacement is the standard case. The chance of an ace is 4/52 = 1/13 for the first card, but 3/51 for the second card if the first was an ace. Two aces in a row therefore have probability 4/52 × 3/51 = 1/221 ≈ 0.004525, while two aces with the first card put back have probability 1/13 × 1/13 = 1/169 ≈ 0.005917. Use the "Dependent events, one after another" mode of the multiple events calculator for such chains. The difference between sampling with and without replacement also separates the binomial and hypergeometric distributions; see binomial vs Poisson vs hypergeometric.

Common mistakes

• Calling events independent because they do not influence each other physically. Independence is a statement about probabilities: check that P(A and B) = P(A) × P(B), or that P(A given B) = P(A).
• Adding the probabilities of events that can happen together. P(A) + P(B) counts the overlap twice unless the events are mutually exclusive; subtract P(A and B). For a king or a heart, 1/13 + 1/4 = 17/52 is too large by 1/52.
• Multiplying the probabilities of mutually exclusive events. P(A) × P(B) is not P(A and B) here: for a king and a queen it would give 1/169 when the true value is 0.
• Treating draws without replacement as independent. The second draw depends on the first, as the two aces show.
• Believing that mutually exclusive events are independent because they are unrelated. They are the most strongly related pair possible: one happening rules the other out.