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 you 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   =   1
52 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 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:

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

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)

 

 
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