Special 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
P (A) =
Count of outcomes in Sample Space S
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.
Count of outcomes in which A happens
Count of outcomes in which A doesn't happen
 Odds of A happening =
P (A)
P (Ac )
 Odds against A happening =
P (Ac )
P (A)
Combinatorics
 When 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
Factorials
 In 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.
Permutations
 A 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:
n!
(n - k )!
 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.
Another way to Count
 Suppose 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 ´ n ´ n ... ´ n = n k
 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.
Combinations
 A 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:
n!
k !(n - k )!
 There are combinations and permutations functions in
your calculator.
Another Way to Count
 Suppose 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:
(n + k - 1)!
k !(n - 1)!
 Remember this refers to “n things taken k at a time”.
Summary of the 4 Ways to Count
Homework
 Worksheet Section 8.3 Day 1.