Probability+-+Complement

=Probability: Complement= Complement of an Event: All outcomes that are **NOT** the event. So the Complement of an event is all the **other** outcomes (**not** the ones you want). And together the Event and its Complement make all possible outcomes.
 * [[image:http://www.mathsisfun.com/data/images/head-tails.jpg width="81" height="70" caption="pair of dice"]] || When the event is **Heads**, the complement is **Tails** ||
 * [[image:http://www.mathsisfun.com/data/images/cards.jpg width="100" height="78"]] || When the event is **{Hearts}** the complement is **{Spades, Clubs, Diamonds, Jokers}** ||
 * || When the event is **{Monday, Wednesday}** the complement is **{Tuesday, Thursday, Friday, Saturday, Sunday}** ||

Probability

 * **Probability of an event** happening = ||  || Number of ways it can happen ||
 * ^  ||^   || Total number of outcomes ||
 * ^  ||^   || Total number of outcomes ||

Example: the chances of rolling a "4" with a die
The probability of an event is shown using "P": The complement is shown by a little ' mark such as A' (or sometimes Ac or A ): The two probabilities always add to 1 P(A) + P(A') = 1
 * Number of ways it can happen: 1** (there is only 1 face with a "4" on it)
 * Total number of outcomes: 6** (there are 6 faces altogether)
 * So the probability = || 1 ||
 * ^  || 6 ||
 * ^  || 6 ||
 * P(A)** means "Probability of Event A"
 * P(A')** means "Probability of the complement of Event A"



Example: Rolling a "5" or "6"
Number of ways it can happen: 2 Total number of outcomes: 6
 * Event A**: {5, 6}
 * P(A) = || 2 || = || 1 ||
 * ^  || 6 ||^   || 3 ||
 * ^  || 6 ||^   || 3 ||

The **Complement of Event A** is {1, 2, 3, 4} Number of ways it can happen: 4 Total number of outcomes: 6 Let us add them: Yep, that makes 1 It makes sense, right? **Event A** plus all outcomes that are **not Event A** make up all possible outcomes.
 * P(A') = || 4 || = || 2 ||
 * ^  || 6 ||^   || 3 ||
 * ^  || 6 ||^   || 3 ||
 * P(A) + P(A') = || 1 || + || 2 || = || 3 || = 1 ||
 * ^  || 3 ||^   || 3 ||^   || 3 ||^   ||
 * ^  || 3 ||^   || 3 ||^   || 3 ||^   ||

Why is the Complement Useful?
It is sometimes easier to work out the complement first.

Example. Throw two dice. What is the probability the two scores are **different**?
Different scores are like getting a **2 and 3**, or a **6 and 1**. It is quite a long list: A = { (1,2), (1,3), (1,4), (1,5), (1,6),

(2,1), (2,3), (2,4), ... etc ! }

But the complement (which is when the two scores are the same) is only **6 outcomes**: A' = { (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) } And the probability is easy to work out: P(A') = 6/36 = **1/6**

Knowing that P(A) and P(A') together make 1, we can calculate: P(A) = 1 - P(A') = 1 - 1/6 = **5/6**

So in this case it's easier to work out P(A') first, then find P(A)

=Reference=

@http://www.mathsisfun.com