Section 8.3 – Equally Likely Outcomes and Ways to Count

EventsSpecial Topics

Calculating Outcomes for Equally Likely Events

If a random phenomenon has equally likely outcomes, then the probability of event A is:

Count of outcomes in event A( ) Count of outcomes in Sample Space SP A =

How to Calculate “Odds”Odds are different from probability, and don’t

follow the rules for probability. They are often used, so they are included here.

Odds of A happening =

Odds against A happening =

Count of outcomes in which A happensCount of outcomes in which A doesn't happen

( )( )cP AP A

( )( )

cP AP A

CombinatoricsWhen outcomes are equally likely, we find

probabilities by counting outcomes. The study of counting methods is called combinatorics.

Combinatorics is the study of methods for counting.

Recall the Fundamental Counting Rule:If you have “m” things of one kind and “n”

things of a second kind, and “p” things of a third kind, then the total combinations of these three things is: m x n x p

FactorialsIn order to study combinatorics, it is

necessary to understand factorials.For a positive integer n, “n factorial” is

notated n! and equals the product of the first n positive integers: n × (n − 1) × (n − 2) × … × 3 × 2 × 1.

By convention, we define 0! to equal 1, not 0, which can be interpreted as saying there is one way to arrange zero items.

PermutationsA permutation is an ordered arrangement of k items that are chosen without replacement from a collection of n items. It can be notated as P(n, k), nPk and and has the formula:

When a problem ask something like how many ways can you arrange 10 books 4 at a time, this is permutations.

Keep in mind that no element in a permutati0n can be repeated.

( )!

!n

n k-

Another way to CountSuppose we have a collection of n distinct

items. We want to arrange k of these items in order, and the same item can appear several times in the arrangement. The number of possible arrangements is:

n is multiplied by itself k times This is related to permutations, but in a

permutation, the item can appear in the arrangement only once.

... kn n n n n´ ´ ´ =

CombinationsA combination is an unordered arrangement

of k items that are chosen without replacement from a collection of n items. It is notated as C(n, k), nCk, or “n choose k”.

The formula for Combinations is:

There are combinations and permutations functions in your calculator.

!!( )!n

k n k-

Another Way to CountSuppose we have a collection of n distinct

items. We want to select k of those items with no regard to order, and any item can appear more than once in the collection. The number of possible collections is:

Remember this refers to “n things taken k at a time”.

( )( )

1 !! 1 !

n kk n

+ --

Summary of the 4 Ways to Count

HomeworkWorksheet Section 8.3 Day 1.