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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.