Mutually Exclusive Events

If two events are mutually exclusive, it simply means that if one happens there is no chance the other can happen at the same time.

‘Heads’ and ‘tails’ on a coin are mutually exclusive – heads and tails cannot occur at the same time (with one coin, anyway).

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But ‘clouds’ and ‘rain’ are not mutually exclusive – they can both occur at the same time.

This can be used in some problems. For instance, if I roll a die what is the probability that it is not a five?

Rolling a 1, 2, 3, 4, or 6 is mutually exclusive to rolling a 5 – if I roll a 5 I can’t have the other numbers at the same time.  I can use this information to solve the question:

Instead of working out the probability of a 1, 2, 3, 4, or 6 occurring, I can just work out the probability of a 5 occurring –

Then because I want to know the probability that it won’t occur, and because I know rolling a 5 is mutually exclusive to rolling any other numbers, I can take 1/6 away from 1 to get 5/6 as my answer.

Picking card at random question

Say I pick a card at random from a pack.  What is the probability that it is not an ace?


The card will either be ‘an ace’ or ‘not an ace’.  These are mutually exclusive outcomes. There are 52 cards in a pack, with 4 of them being aces.

Therefore, the probability of an ace occurring is:


Since the two outcomes ‘an ace’ and ‘not an ace’ are the only two possible outcomes, and they are mutually exclusive, their probabilities must add to 1.

So the probability of “not an ace” must be:


When two outcomes represent all the possible outcomes, and are mutually exclusive, they are complementary events. For example, heads and tails are complementary events.