sst5e_ppt_ch05.pptx

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STATISTICS

INFORMED DECISIONS USING DATA

Fifth Edition, Global Edition

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5.1 Probability Rules

Learning Objectives

1. Apply the rules of probabilities

2. Compute and interpret probabilities using the empirical

method

3. Compute and interpret probabilities using the classical

method

4. Use simulation to obtain data based on probabilities

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5.1 Probability Rules

Introduction to Probability

(1 of 5)

Probability

is a measure of the likelihood of a random

phenomenon or chance behavior occurring. Probability

describes the long-term proportion with which a certain

outcome

will occur in situations with short-term uncertainty.

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5.1 Probability Rules

Introduction to Probability

(2 of 5)

Probability deals with experiments that yield random

short-term results or

outcomes

, yet reveal long-term predictability.

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5.1 Probability Rules

Introduction to Probability

(3 of 5)

The Law of Large Numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets

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5.1 Probability Rules

Introduction to Probability

(4 of 5)

In probability, an

experiment

is any process that can be

repeated in which the results are uncertain.

The

sample space,

S

, of a probability experiment is the

collection of all possible outcomes.

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5.1 Probability Rules

Introduction to Probability

(5 of 5)

EXAMPLE Identifying Events and the Sample Space of a

Probability Experiment

Consider the probability experiment of having two children.

(a) Identify the outcomes of the probability experiment.

(b) Determine the sample space.

(c) Define the event E = “have one boy”.

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5.1 Probability Rules

5.1.1 Apply the Rules of Probabilities

(1 of 4)

Rules of probabilities

1. The probability of any event E, P(E),must be greater than or equal to 0 and less than or equal to 1.

That is, 0 ≤ P(E) ≤ 1.

2. The sum of the probabilities of all outcomes must equal 1.

That is, if the sample space

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5.1 Probability Rules

5.1.1 Apply the Rules of Probabilities

(2 of 4)

A

probability model

lists the possible outcomes of a

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5.1 Probability Rules

5.1.1 Apply the Rules of Probabilities

(3 of 4)

EXAMPLE A Probability Model

In a bag of peanut M&M milk

chocolate candies, the colors of the candies can be brown, yellow, red, blue, orange, or green. Suppose that a candy is randomly selected from a bag. The table shows each color and the probability of drawing that color. Verify this is a probability model.

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5.1 Probability Rules

5.1.1 Apply the Rules of Probabilities

(4 of 4)

If an event is

impossible

, the probability of the event is 0.

If an event is a

certainty,

the probability of the event is 1.

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5.1 Probability Rules

5.1.2 Compute and Interpret Probabilities Using the Empirical Method (1 of 6)

Approximating Probabilities Using the Empirical

Approach

The probability of an event E is approximately the number of times event E is observed divided by the number of repetitions of the

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5.1 Probability Rules

EXAMPLE Building a

Probability Model

Pass the PigsTM is a

Milton-Bradley game in which pigs are used as dice. Points are earned based on the way the pig lands. There are six possible outcomes when one pig is tossed. A class of 52 students rolled pigs 3,939 times. The number of times

Outcome Frequency

Side with no dot 1344

Side with dot 1294

Razorback 767

Trotter 365

Snouter 137

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5.1 Probability Rules

EXAMPLE Building a

Probability Model

(a) Use the results of the experiment to build a probability model for the way the pig lands.

(b) Estimate the probability that a thrown pig lands on the “side with dot”.

(c) Would it be unusual to throw a “Leaning Jowler”?

Outcome Frequency

Side with no dot 1344

Side with dot 1294

Razorback 767

Trotter 365

Snouter 137

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5.1 Probability Rules

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5.1 Probability Rules

Outcome Probability

Side with no dot 0.341

Side with dot 0.329

Razorback 0.195

Trotter 0.093

Snouter 0.035

Leaning Jowler 0.008

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5.1 Probability Rules

Outcome Probability

Side with no dot 0.341

Side with dot 0.329

Razorback 0.195

Trotter 0.093

Snouter 0.035

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5.1 Probability Rules

5.1.3 Compute and Interpret Probabilities Using the Classical

Method

(1 of 5)

The classical method of computing probabilities requires

equally likely outcomes.

An experiment is said to have

equally likely outcomes

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5.1 Probability Rules

5.1.3 Compute and Interpret Probabilities Using the Classical

Method

(2 of 5)

Computing Probability Using the Classical Method

If an experiment has n equally likely outcomes and if the number of ways that an event E can occur is m, then the probability of E,

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5.1 Probability Rules

5.1.3 Compute and Interpret Probabilities Using the Classical

Method

(3 of 5)

Computing Probability Using the Classical Method

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5.1 Probability Rules

5.1.3 Compute and Interpret Probabilities Using the Classical

Method

(4 of 5)

EXAMPLE Computing Probabilities Using the Classical

Method

Suppose a “fun size” bag of M&Ms contains 9 brown candies, 6 yellow candies, 7 red candies, 4 orange candies, 2 blue candies, and 2 green candies. Suppose that a candy is randomly selected.

(a) What is the probability that it is yellow?

(b) What is the probability that it is blue?

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5.1 Probability Rules

5.1.3 Compute and Interpret Probabilities Using the Classical

Method

(5 of 5)

EXAMPLE Computing Probabilities Using the Classical

Method

(a) There are a total of 9 + 6 + 7 + 4 + 2 + 2 = 30 candies, so

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5.1 Probability Rules

5.1.4 Use Simulation to Obtain Data Based on Probabilities

EXAMPLE Using Simulation

Use the probability applet to simulate throwing a 6-sided die 100 times. Approximate the probability of rolling a 4. How does this

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5.1 Probability Rules

5.1.5 Recognize and Interpret Subjective Probabilities

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The subjective probability of an outcome is a probability

obtained on the basis of personal judgment.

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5.1 Probability Rules

5.1.5 Recognize and Interpret Subjective Probabilities

(2 of 2)

EXAMPLE Empirical, Classical, or Subjective Probability

In his fall 1998 article in Chance Magazine, (“A Statistician Reads the Sports Pages,” pp. 17-21,) Hal Stern investigated the

probabilities that a particular horse will win a race. He reports that these probabilities are based on the amount of money bet on each horse. When a probability is given that a particular horse will win a race, is this empirical, classical, or subjective probability?

Subjective because it is based upon people’s feelings about which horse will win the race. The probability is not based on a

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5.2 The Addition Rule and

Complements

Learning Objectives

1. Use the Addition Rule for Disjoint Events

2. Use the General Addition Rule

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

(1 of 8)

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

(2 of 8)

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5.2 The Addition Rule and Complements

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

(4 of 8)

Addition Rule for Disjoint Events

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

(5 of 8)

The Addition Rule for Disjoint Events can be extended to

more than two disjoint events.

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

(6 of 8)

EXAMPLE The Addition

Rule for Disjoint Events

The probability model to the right shows the distribution of the number of rooms in

housing units in the United States.

(a) Verify that this is a probability model.

All probabilities are between 0 and 1, inclusive.

0.010 + 0.032 + … + 0.080 = 1

Number of Rooms

in Housing Unit Probability

One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

(7 of 8)

EXAMPLE The Addition

Rule for Disjoint Events

(b) What is the probability a randomly selected housing unit has two or three rooms?

P(two or three)

= P(two) + P(three) = 0.032 + 0.093

= 0.125

Number of Rooms

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5.2 The Addition Rule and Complements

5.2.1 Use the Addition Rule for Disjoint Events

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EXAMPLE The Addition

Rule for Disjoint Events

(c) What is the probability a randomly selected housing unit has one or two or three rooms?

P(one or two or three)

= P(one) + P(two) + P(three) = 0.010 + 0.032 + 0.093

= 0.135

Number of Rooms

One 0.010 Two 0.032 Three 0.093 Four 0.176 Five 0.219 Six 0.189 Seven 0.122 Eight 0.079

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5.2 The Addition Rule and Complements

5.2.2 Use the General Addition Rule

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The General Addition Rule

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5.2 The Addition Rule and Complements

5.2.2 Use the General Addition Rule

(2 of 3)

EXAMPLE Illustrating the General Addition Rule

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5.2 The Addition Rule and Complements

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5.2 The Addition Rule and Complements

5.2.3 Compute the Probability of an Event Using the

Complement Rule

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Complement of an Event

Let S denote the sample space of a probability experiment and let

E denote an event. The complement of E, denoted EC, is all

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5.2 The Addition Rule and Complements

5.2.3 Compute the Probability of an Event Using the

Complement Rule

(2 of 6)

Complement Rule

If E represents any event and EC_{represents the complement of}_E_,

then

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5.2 The Addition Rule and Complements

5.2.3 Compute the Probability of an Event Using the

Complement Rule

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EXAMPLE Illustrating the Complement Rule

According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog?

P(do not own a dog) = 1 − P(own a dog) = 1 − 0.316

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5.2 The Addition Rule and Complements

5.2.3 Compute the Probability of an Event Using the

Complement Rule

(4 of 6)

The data to the right represent the travel time to work for

residents of Hartford County, CT.

(a) What is the probability a randomly selected resident has a travel time of 90 or more minutes?

Travel Time Frequency

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5.2 The Addition Rule and Complements

5.2.3 Compute the Probability of an Event Using the

Complement Rule

(5 of 6)

Travel Time Frequency

Less than 5 minutes 24,358 5 to 9 minutes 39,112 10 to 14 minutes 62,124 15 to 19 minutes 72,854 20 to 24 minutes 74,386 25 to 29 minutes 30,099 30 to 34 minutes 45,043 35 to 39 minutes 11,169 40 to 44 minutes 8,045 45 to 59 minutes 15,650 60 to 89 minutes 5,451 90 or more minutes 4,895 Source: United States Census Bureau

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5.2 The Addition Rule and Complements

5.2.3 Compute the Probability of an Event Using the

Complement Rule

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(b) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes.

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5.3 Independence and the Multiplication Rule

Learning Objectives

1. Identify independent events

2. Use the Multiplication Rule for Independent Events

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5.3 Independence and the Multiplication Rule

5.3.1 Identify independent events

(1 of 2)

Two events

E

and

F

are

independent

if the occurrence of

event

E

in a probability experiment does not affect the

probability of event

F

. Two events are

dependent

if the

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5.3 Independence and the Multiplication Rule

5.3.1 Identify independent events

(2 of 2)

EXAMPLE Independent or Not?

(a) Suppose you draw a card from a standard 52-card deck of cards and then roll a die. The events “draw a heart” and “roll an even number” are independent because the results of choosing a card do not impact the results of the die toss.

(b) Suppose two 40-year old women who live in the United States are randomly selected. The events “woman 1 survives the

year” and “woman 2 survives the year” are independent.

(c) Suppose two 40-year old women live in the same apartment complex. The events “woman 1 survives the year” and

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5.3 Independence and the Multiplication Rule

5.3.2 Use the Multiplication Rule for Independent Events

(1 of 6)

Multiplication Rule for Independent Events

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5.3 Independence and the Multiplication Rule

5.3.2 Use the Multiplication Rule for Independent Events

(2 of 6)

EXAMPLE Computing Probabilities of Independent Events

The probability that a randomly selected female aged 60 years old will survive the year is 99.186% according to the National Vital

Statistics Report, Vol. 47, No. 28. What is the probability that two randomly selected 60 year old females will survive the year?

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5.3 Independence and the Multiplication Rule

5.3.2 Use the Multiplication Rule for Independent Events

(3 of 6)

EXAMPLE Computing Probabilities of Independent Events

A manufacturer of exercise equipment knows that 10% of their products are defective. They also know that only 30% of their

customers will actually use the equipment in the first year after it is purchased. If there is a one-year warranty on the equipment, what proportion of the customers will actually make a valid warranty

claim?

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5.3 Independence and the Multiplication Rule

5.3.2 Use the Multiplication Rule for Independent Events

(4 of 6)

Multiplication Rule for

n

Independent Events

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5.3 Independence and the Multiplication Rule

5.3.2 Use the Multiplication Rule for Independent Events

(5 of 6)

EXAMPLE Illustrating the Multiplication Principle for

Independent Events

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5.3 Independence and the Multiplication Rule

5.3.2 Use the Multiplication Rule for Independent Events

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EXAMPLE Illustrating the Multiplication Principle for

Independent Events

P(all 4 survive)

= P(1st survives and 2nd survives and 3rd survives and 4th survives)

= P(1st survives) • P(2nd survives) • P(3rd survives) • P(4th survives)

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5.3 Independence and the Multiplication Rule

5.3.3 Compute At-Least Probabilities

(1 of 4)

EXAMPLE Computing “at least” Probabilities

Statistics Report, Vol. 47, No. 28. What is the probability that at least one of 500 randomly selected 60 year old females will die during the course of the year?

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5.3 Independence and the Multiplication Rule

5.3.3 Compute At-Least Probabilities

(2 of 4)

Summary: Rules of Probability

1. The probability of any event must be between 0 and 1. If we let

E denote any event, then 0 ≤ P(E) ≤ 1.

2. The sum of the probabilities of all outcomes in the sample space must equal 1.

That is, if the sample space S = {e₁, e₂, …, e_n}, then P(e₁) +

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5.3 Independence and the Multiplication Rule

5.3.3 Compute At-Least Probabilities

(3 of 4)

Summary: Rules of Probability

3. If E and F are disjoint events, then

P(E or F) = P(E) + P(F).

If E and F are not disjoint events, then

P(E or F) = P(E) + P(F) − P(E and F).

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5.3 Independence and the Multiplication Rule

5.3.3 Compute At-Least Probabilities

(4 of 4)

Summary: Rules of Probability

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5.4 Conditional Probability and the General

Multiplication Rule

Learning Objectives

1. Compute conditional probabilities

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(1 of 8)

Conditional Probability

The notation P(F|E) is read “the probability of event F given event

E”. It is the probability that the event F occurs given that event E

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(2 of 8)

EXAMPLE An Introduction to Conditional Probability

Suppose that a single six-sided die is rolled. What is the

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(3 of 8)

The probability of event F occurring, given the occurrence of event

E, is found by dividing the probability of E and F by the probability of E, or by dividing the number of outcomes in E and F by the

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(4 of 8)

EXAMPLE Conditional Probabilities on Belief about God and Region of the Country

A survey was conducted by the Gallup Organization conducted May 8 − 11, 2008 in which 1,017 adult Americans were asked, “Which of the following statements comes closest to your belief about God – you believe in God, you don’t believe in God, but you do believe in a universal spirit or higher power, or you don’t

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(5 of 8)

blank Believe in

God universal spiritBelieve in Don’t believe in either

East 204 36 15

Midwest 212 29 13

South 219 26 9

West 152 76 26

(a) What is the probability that a randomly selected adult American who lives in the East believes in God?

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(6 of 8)

East 204 36 15

Midwest 212 29 13

South 219 26 9

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(7 of 8)

East 204 36 15

Midwest 212 29 13

South 219 26 9

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.1 Compute Conditional Probabilities

(8 of 8)

EXAMPLE Murder Victims

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.2 Compute Probabilities Using the General Multiplication Rule (1 of 2)

In words, the probability of E and F is the probability of event E

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5.4 Conditional Probability and the General

Multiplication Rule

5.4.2 Compute Probabilities Using the General Multiplication Rule (2 of 2)

EXAMPLE Murder Victims

In 2005, 19.1% of all murder victims were between the ages of 20 and 24 years old. Also in 2005, 86.9% of murder victims were

male given that the victim was 20 − 24 years old. What is the

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5.1 Counting Techniques

Learning Objectives

1. Solve counting problems using the Multiplication rule

2. Solve counting problems using permutations

3. Solve counting problems using combinations

4. Solve counting problems involving permutations with

nondistinct items

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5.5 Counting Techniques

5.5.1 Solve Counting Problems Using the Multiplication

Rule

(1 of 4)

Multiplication Rule of Counting

If a task consists of a sequence of choices in which there are p

selections for the first choice, q selections for the second choice, r

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5.5 Counting Techniques

5.5.1 Solve Counting Problems Using the Multiplication

Rule

(2 of 4)

EXAMPLE Counting the Number of Possible Meals

The fixed-price dinner at Mabenka Restaurant provides the following choices:

Appetizer: soup or salad

Entrée: baked chicken, broiled beef patty, baby beef liver, or roast beef au jus

Dessert: ice cream or cheesecake

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5.5 Counting Techniques

5.5.1 Solve Counting Problems Using the Multiplication

Rule

(3 of 4)

EXAMPLE Counting the Number of Possible Meals

Ordering such a meal requires three separate decisions. Choose an appetizer. For each choice of appetizer, we have 4 choices of entrée, and for each of these 2 • 4 = 8 parings, there are 2 choices for dessert. A total of

2 • 4 • 2 = 16

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5.5 Counting Techniques

5.5.1 Solve Counting Problems Using the Multiplication

Rule

(4 of 4)

If

n

≥ 0 is an integer, the

factorial symbol

,

n

!, is defined as

follows:

n

! =

n

(

n

− 1) • • • • • 3 • 2 • 1

0! = 1

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5.5 Counting Techniques

5.5.2 Solve Counting Problems using Permutations

(1 of 3)

A

permutation

is an ordered arrangement in which

r

objects

are chosen from

n

distinct (different) objects so that

r

≤

n

and

repetition is not allowed. The symbol

_n

P

_r

represents the

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5.5 Counting Techniques

5.5.2 Solve Counting Problems using Permutations

(2 of 3)

Number of Permutations of

n

Distinct Objects Taken

r

at

a Time

The number of arrangements of r objects chosen from n objects, in which

1. the n objects are distinct,

2. repetition of objects is not allowed, and

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5.5 Counting Techniques

5.5.2 Solve Counting Problems using Permutations

(3 of 3)

EXAMPLE Betting on the Trifecta

In how many ways can horses in a 10-horse race finish first, second, and third?

The 10 horses are distinct. Once a horse crosses the finish line, that horse will not cross the finish line again, and, in a race, order is important. We have a permutation of 10 objects taken 3 at a time.

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5.5 Counting Techniques

5.5.3 Solve Counting Problems Using Combinations

(1 of 3)

A

combination

is a collection, without regard to order, in

which

r

objects are chosen from

n

distinct objects with

r

≤

n

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5.5 Counting Techniques

5.5.3 Solve Counting Problems Using Combinations

(2 of 3)

Number of Combinations of

n

Distinct Objects Taken

r

at

a Time

The number of different arrangements of r objects chosen from n

objects, in which

1. the n objects are distinct,

2. repetition of objects is not allowed, and

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5.5 Counting Techniques

5.5.3 Solve Counting Problems Using Combinations

(3 of 3)

EXAMPLE Simple Random Samples

How many different simple random samples of size 4 can be obtained from a population whose size is 20?

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5.5 Counting Techniques

5.5.4 Solve Counting Problems Involving Permutations with

Nondistinct Items

(1 of 2)

Permutations with Nondistinct Items

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5.5 Counting Techniques

5.5.4 Solve Counting Problems Involving Permutations with

Nondistinct Items

(2 of 2)

EXAMPLE Arranging Flags

How many different vertical arrangements are there of 10 flags if 5 are white, 3 are blue, and 2 are red?

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5.5 Counting Techniques

5.5.5 Compute Probabilities Involving Permutations and

Combinations

(1 of 3)

EXAMPLE Winning the Lottery

In the Illinois Lottery, an urn contains balls numbered 1 to 52.

From this urn, six balls are randomly chosen without replacement. For a $1 bet, a player chooses two sets of six numbers. To win, all six numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the