Events, Permutations, and Combinations

Event

An occurrence where there are several outcomes and only 1 will occur. Can be made up of smaller events.

Disjoint Events

Events where exactly one of them occurs like drawing a 10 or a face card.

Addition Rule

For disjointed events. The total number of outcomes is the sum of all events.

Independent Events

The outcomes of one do not affect the outcomes of the others like flipping a coin and rolling a die.

Multiplication Rule

For independent events. The total number of outcomes is the product of all the unique outcomes for each event.

Subtraction Rule

When counting outcomes, you can count the opposite of what you want and subtract:

Roll two distinct dice and want to know how many outcomes don’t add to 10.

There are 6 \* 6 total outcomes, and (5, 5) (4, 6) (6, 4) add to 10.

36 - 3 = 33

Permutations

There are n items and we pick k of them. We then arrange them in a specific order. How many outcomes are there?

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There are 7 items and 3 positions:

For position 1 there are 7 options

For position 2 there are 6 options

For position 3 there are 5 options

7 *** 6 ** 5*

Permutation Formula

To permute k items out of n total, you get:

nPk = P(n, k) = n! / (n - k)!

Combinations

If we have n items and choose k, if order doesn’t matter how many possibilities are there?

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There are 5 items and we choose 3:

There are P(5, 3) = 60 possibilities, but some groups are the same:

(1, 2, 3) = (3, 1, 2)…

They are grouped into groups that are the size of 3!

So you get 60 / 3! = 10

Combination Formula

The number of ways to choose k items out of n total is:

nCk = C(n, k) = n! / (n - k)! k!