Mutually Exclusive Events

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

Examples:

• Turning left and turning right are Mutually Exclusive (you can't do both at the same time)
• Tossing a coin: Heads and Tails are Mutually Exclusive
• Cards: Kings and Aces are Mutually Exclusive

What is not Mutually Exclusive:

• Turning left and scratching your head can happen at the same time
• Kings and Hearts, because we can have a King of Hearts!

Like here:

Probability

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

 

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)"

 

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"

So, we have:

• P(King and Queen) = 0
• P(King or Queen) = (1/13) + (1/13) = 2/13

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:

• P(King ∩ Queen) = 0
• P(King ∪ Queen) = (1/13) + (1/13) = 2/13

Remembering

To help you remember, think:

Not Mutually Exclusive

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

Example: Hearts and Kings

Hearts and Kings together is only the King of Hearts:

But Hearts or Kings is:

• all the Hearts (13 of them)
• all the Kings (4 of them)

But that counts the King of Hearts twice!

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

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:

• people studying French Only must be 16-b
• people studying Spanish Only must be 21-b

And we get:

And we know there are 30 people, so:

And we can put in the correct numbers:

So we know all this now:

• P(French) = 16/30
  
• P(Spanish) = 21/30
  
• P(French Only) = 9/30
  
• P(Spanish Only) = 14/30
  
• P(French or Spanish) = 30/30 = 1
  
• P(French and Spanish) = 7/30

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!

 

Summary:

Mutually Exclusive

A and B together is impossible: P(A and B) = 0
A or B is the sum of A and B: P(A or B) = P(A) + P(B)

Not Mutually Exclusive

Symbols