Probability+-+Mutually+Exclusive+Events

=Mutually Exclusive Events= Examples: What is **not** Mutually Exclusive: Like here: (can't be both) ||  || Hearts and Kings are (can be both) ||
 * Mutually Exclusive**: can't happen at the same time.
 * 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
 * Turning left and scratching your head can happen at the same time
 * Kings and Hearts, because you can have a King of Hearts!
 * [[image:http://www.mathsisfun.com/data/images/set-aces-kings.gif width="211" height="140"]] ||  || [[image:http://www.mathsisfun.com/data/images/set-hearts-kings.gif width="187" height="140"]] ||
 * Aces and Kings are
 * Mutually Exclusive**
 * not** Mutually Exclusive

Probability
Let's look at the probabilities of Mutually Exclusive events. But first, a definition: Example: there are 4 Kings in a deck of 52 cards. What is the probability of picking a King?
 * Probability of an event happening = ||  || Number of ways it can happen ||
 * ^  ||^   || Total number of outcomes ||
 * ^  ||^   || Total number of outcomes ||
 * 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 ||
 * ^  || 52 ||^   || 13 ||

Mutually Exclusive
When two events (call them "A" and "B") are Mutually Exclusive it is **impossible** for them to happen together: //"The probability of A and B together equals 0 (impossible)"// But the probability of A **or** B is the sum of the individual probabilities: //"The probability of A **or** B equals the probability of A **plus** the probability of B"//
 * P(A and B) = 0**
 * P(A or B) = P(A) + P(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: Which is written like this: P(King and Ace) = 0 P(King or Ace) = (1/13) + (1/13) = 2/13
 * 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**

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: Then:
 * scoring no goals (Event "A") is **20%**
 * scoring exactly 1 goal (Event "B") is **15%**
 * 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**"
 * ∪** 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
But Hearts **or** Kings is: 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: //"The probability of A **or** B equals the probability of A **plus** the probability of B//
 * Hearts **and** Kings together is only the King of Hearts: || [[image:http://www.mathsisfun.com/data/images/set-hearts-kings-union.gif width="42" height="82"]] ||
 * all the Hearts (13 of them)
 * all the Kings (4 of them)
 * But that counts the King of Hearts twice!**
 * P(A or B) = P(A) + P(B) - P(A and 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: 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: Lastly, let's check with our formula: Put the values in: Yes, it works!
 * people studying French Only must be 16-b
 * people studying Spanish Only must be 21-b
 * 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
 * P(A or B) = P(A) + P(B) - P(A and B)**
 * 30/30 = 16/30 + 21/30 – 7/30**

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)**

=Reference=

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