Section 8.3 – Equally Likely Outcomes and Ways to Count Events

1. Section 8.3 – Equally Likely Outcomes and Ways to Count Events Special Topics

2. Calculating Outcomes for Equally Likely Events • If a random phenomenon has equally likely outcomes, then the probability of event Ais:

3. 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 =

4. 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

5. Factorials • In order to study combinatorics, it is necessary to understand factorials. • For a positive integer n, “nfactorial” 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.

6. 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), nPkand 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.

7. 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 is multiplied by itself k times • This is related to permutations, but in a permutation, the item can appear in the arrangement only once.

8. 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: • There are combinations and permutations functions in your calculator.

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

10. Summary of the 4 Ways to Count

11. Homework • Worksheet Section 8.3 Day 1.