Probability
Grade 4-5Mutually Exclusive Events
Mutually exclusive events cannot happen at the same time, and that single fact unlocks a shortcut: to find the probability of one event or another, simply add their probabilities together. This lesson covers identifying whether events are mutually exclusive, using the OR rule to add probabilities, and finding a missing probability.
What you need to know
- Mutually exclusive events cannot both happen at the same time.
- For mutually exclusive events, P(A or B) equals P(A) plus P(B).
- If a set of outcomes is exhaustive and mutually exclusive, their probabilities sum to 1.
- Rolling a 2 and rolling an odd number on one die are mutually exclusive.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Identifying mutually exclusive events
Two events are mutually exclusive if they cannot both happen at the same time, meaning there is no outcome that satisfies both events at once. To check, look for any outcome that would count for both events.
Rolling a dice, "scoring a 3" and "scoring an even number" are mutually exclusive, since 3 is not even and no single roll can satisfy both. "Scoring a multiple of 3" and "scoring an even number" are not mutually exclusive, since 6 is both.
Using the OR rule for mutually exclusive events
When events are mutually exclusive, the probability of one event or another happening is found by adding their individual probabilities together. This extends to three or more mutually exclusive events by adding all of them.
Events A and B are mutually exclusive, with P(A) = 0.3 and P(B) = 0.25. The probability of A or B is 0.3 + 0.25 = 0.55.
Finding a missing probability
If P(A or B) and one of the individual probabilities are known, the other can be found by rearranging the OR rule and subtracting.
Events A and B are mutually exclusive, with P(A or B) = 0.7 and P(A) = 0.45. Then P(B) = 0.7 - 0.45 = 0.25.
Applying the OR rule within a single event
Some questions describe two outcomes of the same experiment, such as a dice roll or a random selection, and ask for the probability of one outcome or another. If the outcomes cannot overlap, count or work out each probability separately and add them.
A bag has 12 balls numbered 1 to 12. The probability of picking a multiple of 4 or a multiple of 5 is 3/12 + 2/12 = 5/12, since no number from 1 to 12 is a multiple of both 4 and 5.
Worked examples
Three exam-style questions, fully solved
A fair 6-sided dice is rolled. State, giving a reason, whether the events 'scoring a 3' and 'scoring an even number' are mutually exclusive.
Easy- 1.Check whether any single score could satisfy both events
- 2.Recognise that 3 is not an even number
Answer: Yes, they are mutually exclusive, since both cannot happen on the same roll
Events A and B are mutually exclusive. P(A) = 0.3, P(B) = 0.25. Find P(A or B).
Medium- 1.Recall that for mutually exclusive events, P(A or B) = P(A) + P(B)
- 2.Add the two probabilities: 0.3 + 0.25
Answer: 0.55
A bag has 12 balls numbered 1 to 12. A ball is picked at random. Find the probability that it shows a multiple of 4 or a multiple of 5.
Hard- 1.Count the multiples of 4 from 1 to 12: 4, 8, 12, which is 3 out of 12
- 2.Count the multiples of 5 from 1 to 12: 5, 10, which is 2 out of 12
- 3.Since no number is a multiple of both, add the two probabilities: 3/12 + 2/12
Answer: 5/12
Practice
6 questions, marked instantly
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Practice
Now try these yourself.
Type your answer and check it. The worked solution appears once you have had a go.
Two events are mutually exclusive. P(A) = 0.3 and P(B) = 0.5. Work out P(A or B).
A dice is rolled. Work out P(getting a 2 or getting a 5).
Events A and B are mutually exclusive. P(A) = 1/4 and P(A or B) = 3/4. Work out P(B).
A bag has only red, blue and green counters. P(red) = 0.4 and P(blue) = 0.25. Work out P(green).
A card is drawn from a standard pack. Are "drawing a heart" and "drawing a black card" mutually exclusive? Answer yes or no.
A dice is rolled. Are "getting an even number" and "getting a multiple of 3" mutually exclusive? Answer yes or no.
Avoid these
The mistakes students make most often
Assuming two events are mutually exclusive without checking whether an outcome could satisfy both, for example missing that 6 is both a multiple of 3 and an even number.
Multiplying the individual probabilities instead of adding them when finding P(A or B) for mutually exclusive events.
Adding probabilities directly for events that are not actually mutually exclusive, double-counting any overlap between them.
Rearranging P(A or B) = P(A) + P(B) incorrectly when solving for a missing probability, for example adding instead of subtracting.
Giving a vague reason when asked to justify whether events are mutually exclusive, instead of clearly identifying a shared outcome or explaining why none exists.
FAQ
Questions students and parents ask
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