Mutually Exclusive Events

road fork

Mutually Exclusive: can't happen at the same time.


What is not Mutually Exclusive:

Like here:

set aces kings separate set hearts kings joins at king of hearts
Aces and Kings are
Mutually Exclusive
(can't be both)
 Hearts and Kings are
Mutually Exclusive
(can be both)


Let's look at the probabilities of Mutually Exclusive events. But first, a definition:

Probability of an event happening = Number of ways it can happenTotal number of outcomes


Example: there are 4 Kings in a deck of 52 cards. What is the probability of picking a King?

Number of ways it can happen: 4 (there are 4 Kings)

Total number of outcomes: 52 (there are 52 cards in total)

So the probability = 452 = 113

Mutually Exclusive

When two events (call them "A" and "B") are Mutually Exclusive it is impossible for them to happen together:

P(A and B) = 0

"The probability of A and B together equals 0 (impossible)"

Example: King AND Queen

A card cannot be a King AND a Queen at the same time!


But, for Mutually Exclusive events, the probability of A or B is the sum of the individual probabilities:

P(A or B) = P(A) + P(B)

"The probability of A or B equals the probability of A plus the probability of B"

Example: King OR Queen

In a Deck of 52 Cards:


When we combine those two Events:

Which is written like this:

P(King or Queen) = (1/13) + (1/13) = 2/13

So, we have:

Special Notation

Instead of "and" you will often see the symbol (which is the "Intersection" symbol used in Venn Diagrams)

Instead of "or" you will often see the symbol (the "Union" symbol)

So we can also write:

soccer teams

Example: Scoring Goals

If the probability of:



Which is written:

P(A B) = 0

P(A B) = 20% + 15% = 35%


To help you remember, think:

union cup

"Or has more ... than And"

Also is like a cup which holds more than

Not Mutually Exclusive

Now let's see what happens when events are not Mutually Exclusive.

Example: Hearts and Kings

set hearts kings joins at king of hearts

Hearts and Kings together is only the King of Hearts:

set hearts kings union

But Hearts or Kings is:

But that counts the King of Hearts twice!

So we correct our answer, by subtracting the extra "and" part:

set hearts kings sum

16 Cards = 13 Hearts + 4 Kings − the 1 extra King of Hearts

Count them to make sure this works!

As a formula this is:

P(A or B) = P(A) + P(B) − P(A and B)

"The probability of A or B equals the probability of A plus the probability of B
minus the probability of A and B"

Here is the same formula, but using and :

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

A Final Example

16 people study French, 21 study Spanish and there are 30 altogether. Work out the probabilities!

This is definitely a case of not Mutually Exclusive (you can study French AND Spanish).

Let's say b is how many study both languages:

And we get:

set language ex1

And we know there are 30 people, so:

(16−b) + b + (21−b) = 30
37 − b = 30
b = 7

And we can put in the correct numbers:

set language ex2

So we know all this now:

Lastly, let's check with our formula:

P(A or B) = P(A) + P(B) − P(A and B)

Put the values in:

30/30 = 16/30 + 21/30 − 7/30

Yes, it works!



Mutually Exclusive

Not Mutually Exclusive