Chapter 12 – Probability and Statistics12.1 – The Counting Principle

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?• 2• 5• 6• 9

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?

Period 1st 2nd 3rd 4th 5th 6th

Number of Choices 6 5 4 3 2 1

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

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