# 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 Mutually Exclusive not (can be both) |

## Probability

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

Probability of an event happening = \frac{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 = \frac{4}{52} = \frac{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)"*

But 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: A Deck of Cards

In a Deck of 52 Cards:

- the probability of a King is 1/13, so
**P(King)=1/13** - the probability of an Ace is also 1/13, so
**P(Ace)=1/13**

When we combine those two Events:

- The probability of a card being a King
**and**an Ace is**0**(Impossible) - The probability of a card being a King
**or**an Ace is (1/13) + (1/13) =**2/13**

Which is written like this:

P(King and Ace) = 0

P(King or Ace) = (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)

### 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 |

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)**