Chapter 12 – Probability and Statistics 12.1 – The Counting Principle

Today we will learn how to: Solve problems involving independent and dependent events Solve problems involving dependent events

12.1 – The Counting Principle Outcome – the result of a single trial Flipping a coin – 2 outcomes – heads or tails Sample space – set of all possible outcomes Event – one or more outcomes of a trial Independent events – events that do not affect one another

12.1 – The Counting Principle Example 1 A sandwich menu offers customers a choice of white, wheat, or rye bread with one spread chosen from butter, mustard, or mayonnaise. How many different combinations of bread and spread are possible?

12.1 – The Counting Principle Notice that in Example 1, there are 3 ways to choose the bread, 3 ways to choose the spread, and 3 · 3 or 9 ways to choose a combination of the two. This illustrated the Fundamental Counting Principle

12.1 – The Counting Principle Fundamental Counting Principle If event M can occur in m ways and is followed by event N that can occur in n ways, then event M followed by event N can occur in m · n ways If event M can occur in 2 ways and event N can occur in 3 ways, then M followed by N can occur in 2 · 3 or 6 ways This rule can be extended to any number of events

12.1 – The Counting Principle Example 2 The Murray family is choosing from a trip to the beach or a trip to the mountains. The family can select transportation from a car, train, or plane. How many different ways can the family select a destination followed by a means of transportation?

12.1 – The Counting Principle Example 3 How many iPhone numeric password codes are possible?

12.1 – The Counting Principle Dependent Events – the outcome of one event does affect the outcome of another event The Fundamental Counting Principle applies to dependent events as well

12.1 – The Counting Principle Example 4 How many different schedules could a student who is planning to take only four different classes have? Period1 st 2 nd 3 rd 4 th 5 th 6 th Number of Choices654321

12.1 – The Counting Principle Factorial – if n is a positive integer, then n! = n(n – 1)(n – 2)…2 · 1 ! – symbol for factorial 5! = 5 · 4 · 3 · 2 · 1

12.1 – The Counting Principle Independent Events – If the outcome of an event does not affect the outcome of another event, the two events are independent Tossing a coin and rolling a die are independent events Dependent Events – If the outcome of an event does affect the outcome of another event, the two events are dependent Taking a piece of candy from a jar and then taking a second piece without replacing the first are dependent events

12.1 – The Counting Principle HOMEWORK Page 687 #2 – 28