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:
 |
 |
|
| Aces and Kings are Mutually Exclusive (can't be both) |
Hearts and Kings are not Mutually Exclusive (can be both) |
Probability
Let's look at the probabilities of Mutually Exclusive events. But first, a definition:
Probability of an event happening = Number of ways it can happen Total 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 = 4 52 = 1 13
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!
- The probability of a King and a Queen is 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"
Example: King OR Queen
In a Deck of 52 Cards:
- the probability of a King is 1/13, so P(King)=1/13
- the probability of a Queen is also 1/13, so P(Queen)=1/13
When we combine those two Events:
- The probability of a King or a Queen is (1/13) + (1/13) = 2/13
Which is written like this:
P(King or Queen) = (1/13) + (1/13) = 2/13
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
Example: Scoring Goals
If the probability of:
- scoring no goals (Event "A") is 20%
- scoring exactly 1 goal (Event "B") is 15%
Then:
- The probability of scoring no goals and 1 goal is 0 (Impossible)
- The probability of scoring no goals or 1 goal is 20% + 15% = 35%
Which is written:
P(A ∩ B) = 0
P(A ∪ B) = 20% + 15% = 35%
Remembering
To help you remember, think:
"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
|
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
- A or B is the sum of A and B minus A and B: P(A or B) = P(A) + P(B) − P(A and B)
Symbols
- And is ∩ (the "Intersection" symbol)
- Or is ∪ (the "Union" symbol)