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10.1 – Counting by Systematic Listing

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10.1 – Counting by Systematic Listing

One-Part Tasks

The results for simple, one-part tasks can often be listed easily.

Tossing a fair coin:

Heads or tails

Rolling a single fair die

1, 2, 3, 4, 5, 6

Consider a club N with four members:

N = {Mike, Adam, Ted, Helen}or N = {M, A, T, H}

In how many ways can this group select a president?

There are four possible results:

M, A, T, and H.

10.1 – Counting by Systematic Listing

Product Tables for Two-Part Tasks

Determine the number of two-digit numbers that can be written using the digits from the set {2, 4, 6}.

The task consists of two parts:

1. Choose a first digit

2. Choose a second digit

The results for a two-part task can be pictured in a product table.

2

4

6

2

26

24

22

4

9 possible numbers

42

44

46

6

62

64

66

10.1 – Counting by Systematic Listing

Product Tables for Two-Part Tasks

What are the possible outcomes of rolling two fair die?

10.1 – Counting by Systematic Listing

Product Tables for Two-Part Tasks

Find the number of ways club N can elect a president and secretary.

N = {Mike, Adam, Ted, Helen}or N = {M, A, T, H}

The task consists of two parts:

1. Choose a president2. Choose a secretary

MA

MM

MT

MH

12 outcomes

AA

AT

AM

AH

TH

TM

TA

TT

HM

HA

HT

HH

10.1 – Counting by Systematic Listing

Product Tables for Two-Part Tasks

Find the number of ways club N can elect a two member committee.

N = {Mike, Adam, Ted, Helen}or N = {M, A, T, H}

6 committees

MA

MM

MT

MH

AA

AT

AH

AM

TM

TA

TH

TT

HM

HA

HT

HH

10.1 – Counting by Systematic Listing

Tree Diagrams for Multiple-Part Tasks

A task that has more than two parts is not easy to analyze with a product table. Another helpful device is a tree diagram.

Find the number of three digit numbers that can be written using the digits from the set {2, 4, 6} assuming repeated digits are not allowed.

A product table will not work for more than two digits.

Generating a list could be time consuming and disorganized.

10.1 – Counting by Systematic Listing

Tree Diagrams for Multiple-Part Tasks

Find the number of three digit numbers that can be written using the digits from the set {2, 4, 6} assuming repeated digits are not allowed.

1st #

2nd #

3rd #

246

4

6

2

6

4

264

2

6

426

6 possibilities

4

6

2

462

2

4

624

6

4

2

642

10.1 – Counting by Systematic Listing

Other Systematic Listing Methods

There are additional systematic ways to produce complete listings of possible results besides product tables and tree diagrams.

How many triangles (of any size) are in the figure below?

D

One systematic approach is begin with A, and proceed in alphabetical order to write all 3-letter combinations (like ABC, ABD, …), then cross out ones that are not triangles and those that repeat.

E

C

F

B

A

Another approach is to “chunk” the figure to smaller, more manageable figures.

There are 12 triangles.

10.2 – Using the Fundamental Counting Principle

Uniformity Criterion for Multiple-Part Tasks:

A multiple part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for previous parts.

Uniformity exists:

Find the number of three letter combinations that can be written using the letters from the set {a, b, c} assuming repeated letters are not allowed.

2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate a list of the possible outcomes by drawing a tree diagram.

Uniformity does not exists:

A computer printer allows for optional settings with a panel of three on-off switches. Set up a tree diagram that will show how many setting are possible so that no two adjacent switches can be on?

10.2 – Using the Fundamental Counting Principle

Uniformity

Find the number of three letter combinations that can be written using the letters from the set {a, b, c} assuming repeated letters are not allowed.

1st letter

2nd letter

3rd letter

abc

b

c

a

c

b

acb

a

c

bac

6 possibilities

b

c

a

bca

a

b

cab

c

b

a

cba

10.2 – Using the Fundamental Counting Principle

Uniformity

2 dimes and one six-sided die numbered from 1 to 6 are tossed. Generate a list of the possible outcomes by drawing a tree diagram.

Die #

Dime

1 d1

d1

d1

d1

d1

d1

d1

1

1 d2

d2

d2

d2

d2

d2

d2

2 d1

2

2 d2

3 d1

3

3 d2

12 possibilities

4 d1

4

4 d2

5 d1

5

5d2

6 d1

6

6 d2

10.2 – Using the Fundamental Counting Principle

Uniformity does not exist

A computer printer is designed for optional settings with a panel of three on-off switches. Set up a tree diagram that will show how many setting are possible so that no two adjacent switches can be on? (o = on, f = off)

1st switch

2nd switch

3rd switch

o

o

f

o

o

f

f

o

o

f

f

o

f

f

10.2 – Using the Fundamental Counting Principle

Fundamental Counting Principle

The principle which states that all possible outcomes in a sample space can be found by multiplying the number of ways each event can occur.

Example:

At a firehouse fundraiser dinner, one can choose from 2 proteins (beef and fish), 4 vegetables (beans, broccoli, carrots, and corn), and 2 breads (rolls and biscuits). How many different protein-vegetable-bread selections can she make for dinner?

Proteins Vegetables Breads

2

4

2

=

16 possible selections

10.2 – Using the Fundamental Counting Principle

Example

At the local sub shop, customers have a choice of the following: 3 breads (white, wheat, rye), 4 meats (turkey, ham, chicken, bologna), 6 condiments (none, brown mustard, spicy mustard, honey mustard, ketchup, mayo), and 3 cheeses (none, Swiss, American). How many different sandwiches are possible?

Breads Meats Condiments Cheeses

6

3

3

4

=

216 possible sandwiches

10.2 – Using the Fundamental Counting Principle

Example:

Consider the set of digits: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.

(a) How many two digit numbers can be formed if repetitions are allowed?

1st digit 2nd digit

10

90

9

=

(b) How many two digit numbers can be formed if no repetitions are allowed?

1st digit 2nd digit

9

81

9

=

(c) How many three digit numbers can be formed if no repetitions are allowed?

1st digit 2nd digit 3rd digit

9

9

8

=

648

10.2 – Using the Fundamental Counting Principle

Example:

(a) How many five-digit codes are possible if the first two digits are letters and the last three digits are numerical?

1st digit 2nd digit 3rd digit 4th digit 5th digit

26

10

10

26

10

676000 possible five-digit codes

(a) How many five-digit codes are possible if the first two digits are letters and the last three digits are numerical and repeats are not permitted?

1st digit 2nd digit 3rd digit 4th digit 5th digit

25

10

9

26

8

468000 possible five-digit codes

10.2 – Using the Fundamental Counting Principle

Factorials

For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!.

For any counting number n, the quantity n factorial is calculated by:

n! = n(n – 1)(n – 2)…(2)(1).

Definition of Zero Factorial:

0! = 1

Examples:

b) (4 – 1)!

a) 4!

c)

3!

4321

321

20

=

24

=

54

6

10.2 – Using the Fundamental Counting Principle

Arrangements of Objects

Factorials are used when finding the total number of ways to arrange a given number of distinct objects.

The total number of different ways to arrange n distinct objects is n!.

Example:

How many ways can you line up 6 different books on a shelf?

1

5

4

3

6

2

720 possible arrangements

10.2 – Using the Fundamental Counting Principle

Arrangements of n Objects Containing Look-Alikes

The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by

Example:

Determine the number of distinguishable arrangements of the letters of the word INITIALLY.

9 letters

with 3 I’s

and 2 L’s

9!

30240 possible arrangements

3!

2!

10.3 – Using Permutations and Combinations

Permutation: The number of ways in which a subset of objects can be selected from a given set of objects, where order is important.

Given the set of three letters, {A, B, C}, how many possibilities are there for selecting any two letters where order is important?

(AB, AC, BC, BA, CA, CB)

Combination: The number of ways in which a subset of objects can be selected from a given set of objects, where order is not important.

Given the set of three letters, {A, B, C}, how many possibilities are there for selecting any two letters where order is not important?

(AB, AC, BC).

10.3 – Using Permutations and Combinations

Factorial Formula for Permutations

Factorial Formula for Combinations

10.3 – Using Permutations and Combinations

Evaluate each problem.

d) 6C6

b) 5C3

c) 6P6

a) 5P3

543

60

720

10

1

10.3 – Using Permutations and Combinations

How many ways can you select two letters followed by three digits for an ID if repeats are not allowed?

Two parts:

1. Determine the set of two letters.

2. Determine the set of three digits.

26P2

10P3

2625

1098

650

720

650720

468,000

10.3 – Using Permutations and Combinations

A common form of poker involves hands (sets) of five cards each, dealt from a deck consisting of 52 different cards. How many different 5-card hands are possible?

Hint: Repetitions are not allowed and order is not important.

52C5

2,598,960

5-card hands

10.3 – Using Permutations and Combinations

Find the number of different subsets of size 3 in the set: {m, a, t, h, r, o, c, k, s}.

Find the number of arrangements of size 3 in the set: {m, a, t, h, r, o, c, k, s}.

9C3

9P3

987

504

arrangements

84

Different subsets

10.3 – Using Permutations and Combinations

Guidelines on Which Method to Use

11.1 – Probability – Basic Concepts

Probability

The study of the occurrence of random events or phenomena.

It does not deal with guarantees, but with the likelihood of an occurrence of an event.

Experiment:

- Any observation or measurement of a random phenomenon.

Outcomes:

- The possible results of an experiment.

Sample Space:

- The set of all possible outcomes of an experiment.

Event:

- A particular collection of possible outcomes from a sample space.

11.1 – Probability – Basic Concepts

Example:

If a single fair coin is tossed, what is the probability that it will land heads up?

Sample Space:

S = {h, t}

Event of Interest:

E = {h}

P(heads) = P(E) =

1/2

The probability obtained is theoretical as no coin was actually flipped

Theoretical Probability:

number of favorable outcomes

n(E)

=

P(E) =

total number of outcomes

n(S)

11.1 – Probability – Basic Concepts

Example:

A cup is flipped 100 times. It lands on its side 84 times, on its bottom 6 times, and on its top 10 times. What is the probability that it lands on it top?

10

1

number of top outcomes

=

=

P(top) =

total number of flips

100

10

The probability obtained is experimental or empirical as the cup was actually flipped.

Empirical or Experimental Probability:

number of times event E occurs

P(E)

͌

number of times the experiment was performed

11.1 – Probability – Basic Concepts

Example:

There are 2,598,960 possible five-card hand in poker. If there are 36 possible ways for a straight flush to occur, what is the probability of being dealt a straight flush?

number of possible straight flushes

P(straight flush) =

total number of five-card hands

36

=

=

0.0000139

2,598,960

This probability is theoretical as no cards were dealt.

11.1 – Probability – Basic Concepts

Example:

A school has 820 male students and 835 female students. If a student is selected at random, what is the probability that the student would be a female?

number of possible female students

P(female) =

total number of students

835

167

835

=

=

=

331

1655

820 + 835

0.505

P(female) =

This probability is theoretical as no experiment was performed.

11.1 – Probability – Basic Concepts

The Law of Large Numbers

As an experiment is repeated many times over, the experimental probability of the events will tend closer and closer to the theoretical probability of the events.

Flipping a coin

Spinner

Rolling a die

11.1 – Probability – Basic Concepts

Odds

A comparison of the number of favorable outcomes to the number of unfavorable outcomes.

Odds are used mainly in horse racing, dog racing, lotteries and other gambling games/events.

Odds in Favor: number of favorable outcomes (A) to the number of unfavorable outcomes (B).

A to B

A : B

Example:

What are the odds in favor of rolling a 2 on a fair six-sided die?

1 : 5

What is the probability of rolling a 2 on a fair six-sided die?

1/6

11.1 – Probability – Basic Concepts

Odds

Odds against: number of unfavorable outcomes (B) to the number of favorable outcomes (A).

B to A

B : A

Example:

What are the odds against rolling a 2 on a fair six-sided die?

5 : 1

What is the probability against rolling a 2 on a fair six-sided die?

5/6

11.1 – Probability – Basic Concepts

Odds

Example:

Two hundred tickets were sold for a drawing to win a new television. If you purchased 10 tickets, what are the odds in favor of you winning the television?

10

Favorable outcomes

200 – 10 =

190

Unfavorable outcomes

10 : 190

=

1 : 19

What is the probability of winning the television?

10/200

=

1/20

=

0.05

11.1 – Probability – Basic Concepts

Converting Probability to Odds

Example:

The probability of rain today is 0.43. What are the odds of rain today?

P(rain) = 0.43

Of the 100 total outcomes, 43 are favorable for rain.

Unfavorable outcomes:

100 – 43 =

57

43 : 57

The odds for rain today:

57 : 43

The odds against rain today:

11.1 – Probability – Basic Concepts

Converting Odds to Probability

Example:

The odds of completing a college English course are 16 to 9. What is the probability that a student will complete the course?

16 : 9

The odds for completing the course:

Favorable outcomes + unfavorable outcomes = total outcomes

16 + 9 = 25

P(completing the course) =

= 0.64