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Mutually Exclusive Events

By Vighnesh Hemnani

Probability is the study of chances of events that can happen on the daily, and some of these events are mutually exclusive! Learn to recognize mutually exclusive events, solved examples and practice problems at Thinkster Math.

Why is this concept useful?

Why is the concept useful? Mutually exclusive events are basically a way of describing events that cannot occur together. In real life, this situation occurs more times than you realize. Some other examples can include: You cannot run while you walk You cannot sleep while eating You cannot take a right turn while taking a left turn This concept is useful because it helps us classify or categorize an event so that in problems of calculating probability, we know that P(A∩B) = 0 for any mutually exclusive events A and B.

Where does this concept fit into the curriculum?

High School Statistics & Probability

What are mutually exclusive events?

The term “mutually exclusive” simply means that something cannot happen at the same time as something else. So, when you apply that to an event in probability, mutually exclusive events just mean that there are two (or more) events that cannot happen at the same time. A popular example of mutually exclusive events is tossing a coin: you can either land a heads or a tails but you cannot have both!

How to use this concept?

You can use this concept of mutually exclusive events in probability problems in order to identify which events are mutually exclusive and then imply that P(A∩B) = 0 for those two events (or maybe more). Knowing this aids the process of finding other missing quantities in probability problems (have a look at some of the solved examples below). P(A∩B) is commonly used in many formulae, one of the being:

P(A∪B) = P(A) + P(B) -P(A∩B)

Sample Math Problems

1. Question: Identify if the follow situation entails any mutually exclusive events or not:

You are at a fair and there is a spin-the-wheel contest. Rules are simple: you spin the wheel, whatever the arrow lands on is your prize (equal chance of landing on any section of the wheel). After spinning, you land on either “No Prize” or “Teddy Bear”.

Answer: Mutually exclusive events.

This is because you cannot land on two or more sections on the wheel and win two or more prizes. Only one state can happen at once, thus the events of either landing on “No Prize” or “Teddy Bear” are mutually exclusive events.

2. Question: A magician has a magic hat that contains 7 bunnies: 3 are brown, 2 are white, and 2 are black. As a part of his trick, the magician pulls a bunny out of the hat randomly – what is the probability the bunny he pulled out is white or black?

Answer:

47\frac{4}{7}

Let event W be randomly pulling out a white bunny and B be randomly pulling out a black bunny. Based on the information we know:

P(W) = 27      P(B) = 27P( W) \ =\ \frac{2}{7} \ \ \ \ \ \ P( B) \ =\ \frac{2}{7}

And the fact that these two events are mutually exclusive since you can’t pull out a white and a black bunny together (while only pulling one bunny out) so: P(W∩B) = 0

Now we need to calculate P(W∪B):
P(W∪B) = P(W) + P(B) -P(W∩B) =

27 +27−0=47\frac{2}{7} \ +\frac{2}{7} -0=\frac{4}{7}

3. Question: Given the following information, find out if the two events, A and B, are mutually exclusive:

P(A)=12 P(B) = 310 P(A∪B) = 1215P( A) =\frac{1}{2} \ P( B) \ =\ \frac{3}{10} \ P( A\cup B) \ =\ \frac{12}{15}

Answer: Yes, they are mutually exclusive. We can begin with the formula for unions: P(A∪B) = P(A) + P(B) -P(A∩B)

Plugging in the values that we already know, we will be left with the probability of A and B, so we can solve for it. If it is 0, it is mutually exclusive:

1215=12+310 − P(A∩B)P(A∩B)=12 + 310 − 1215P(A∩B) = 0 \begin{array}{l} \frac{12}{15} =\frac{1}{2} +\frac{3}{10} \ -\ P( A\cap B)\\ \\ P( A\cap B) =\frac{1}{2} \ +\ \frac{3}{10} \ -\ \frac{12}{15}\\ \\ P( A\cap B) \ =\ 0 \end{array}

Thus, events A and B are mutually exclusive




4. Question: Given the following information, find out if the two events, A and B, are mutually exclusive:

P(A) = 720  P(B) = 27   P(A∩B)=59P( A) \ =\ \frac{7}{20} \ \ P( B) \ =\ \frac{2}{7} \ \ \ P( A\cap B) =\frac{5}{9}


Answer: Not mutually exclusive events.

This is actually a very straight-forward question and requires minimal calculation. Usually, when two events are mutually exclusive, P(A∩B)=0 as there are no situations/instances where both events can happen. But in the question, P(A∩B)≠0=59 therefore A and B cannot be mutually exclusive.

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Practice Math Problems

Given the following information, find out if the two events, A and B, are mutually exclusive:

P(A)=35  P(B) = 12  P(A∣B) = 3350P( A) =\frac{3}{5} \ \ P( B) \ =\ \frac{1}{2} \ \ P( A|B) \ =\ \frac{33}{50}

Given the following information, find out if the two events, A and B, are mutually exclusive:

P(A)=1440  P(B) = 2240  P(A∣B) = 0P( A) =\frac{14}{40} \ \ P( B) \ =\ \frac{22}{40} \ \ P( A|B) \ =\ 0

4. If we know that events A and B are mutually exclusive, find the missing

P(A) = 67  P(B) = 512   P(A∪B) = ?P( A) \ =\ \frac{6}{7} \ \ P( B) \ =\ \frac{5}{12} \ \ \ P( A\cup B) \ =\ ?

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