10.1 – Counting by Systematic ListingOne-Part Tasks

The results for simple, one-part tasks can often be listed easily. Tossing a fair coin:

Rolling a single fair dieHeads or tails1, 2, 3, 4, 5, 6

Consider a club N with four members:

There are four possible results:

In how many ways can this group select a president?

M, A, T, and H.

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

10.1 – Counting by Systematic ListingProduct 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 digit2. Choose a second digitThe results for a two-part task can be pictured in a product table.

Second Digit

First Digit

246

2 4 622

4464

244262

264666

9 possible numbers

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

What are the possible outcomes of rolling two fair die?

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

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

The task consists of two parts:1. Choose a president 2. Choose a secretary

SecretaryM A T H

Pres. MATH

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

HHHMTM

AM

MM

AH

MHMTMA

ATAA

THTTTAHTHA

12 outcomes

10.1 – Counting by Systematic ListingProduct Tables for Two-Part Tasks

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

2nd Member

M A T H1st

Member MATH

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

HHHMTM

AM

MM

AH

MHMTMA

ATAA

THTTTAHTHA

6 committees

10.1 – Counting by Systematic ListingTree 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 ListingTree 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.

26

6 246

1st # 2nd # 3rd #

4

6

4

2

6

2

4

4

6

2

4

2

264

426

462

624

642

6 possibilities

10.1 – Counting by Systematic ListingOther Systematic Listing Methods

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

A B

C

D

E

F

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

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.

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

There are 12 triangles.

10.2 – Using the Fundamental Counting PrincipleUniformity 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.

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.

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?

Uniformity exists:

Uniformity does not exists:

Uniformity

ac

c abc

1st letter 2nd letter 3rd letter

b

c

b

a

c

a

b

b

c

a

b

a

acb

bac

bca

cab

cba

6 possibilities

10.2 – Using the Fundamental Counting Principle

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.

Uniformity

1 d1 Die # Dime

12 possibilities

10.2 – Using the Fundamental Counting Principle

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.

2

4

6

1

3

5

d1d2

d1d2

d1d2

d1d2

d1d2

d1d2

1 d2 2 d1 2 d2 3 d1 3 d2 4 d1 4 d2 5 d1 5d2 6 d1 6 d2

Uniformity does not exist

1st switch 2nd switch 3rd switch

f

o

o

f

o

f

o

f

10.2 – Using the Fundamental Counting Principle

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)

o

fo

f

o

f

10.2 – Using the Fundamental Counting PrincipleFundamental 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 PrincipleExampleAt 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 Cheeses3 4 6 =

216 possible sandwiches

3

10.2 – Using the Fundamental Counting PrincipleExample: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?

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

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

1st digit 2nd digit

9 9 81 =1st digit 2nd digit

9 10 90 =

1st digit 2nd digit 3rd digit9 9 8 = 648

10.2 – Using the Fundamental Counting PrincipleExample:(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

676000 possible five-digit codes26 26 10 1010

(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

468000 possible five-digit codes26 25 10 89

10.2 – Using the Fundamental Counting PrincipleFactorials

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).

Examples:a) 4! b) (4 – 1)! c)

4321 24

3!321

6

54 20= =

Definition of Zero Factorial:0! = 1

10.2 – Using the Fundamental Counting Principle

Example:

Arrangements of ObjectsFactorials 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!.

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

6 5 4 23 1

720 possible arrangements

10.2 – Using the Fundamental Counting Principle

Example:

9!3! 2!

30240 possible arrangements

Arrangements of n Objects Containing Look-AlikesThe 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

1 2

! .! ! !k

nn n n

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

9 letters with 3 I’s and 2 L’s

10.3 – Using Permutations and CombinationsPermutation: The number of ways in which a subset of objects can be selected from a given set of objects, where order is important.

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 important?

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

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 CombinationsFactorial Formula for Permutations

! .( )!n r

nPn r

Factorial Formula for Combinations

! .! !( )!

n rn r

P nCr r n r

10.3 – Using Permutations and CombinationsEvaluate each problem.

c) 6P6a) 5P3b) 5C3 d) 6C6

54360

10

720

1

10.3 – Using Permutations and CombinationsHow many ways can you select two letters followed by three digits for an ID if repeats are not allowed? Two parts:

2. Determine the set of three digits.1. Determine the set of two letters.

26P2 10P3

2625650

1098720650720468,000

10.3 – Using Permutations and CombinationsA 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 CombinationsFind 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

84

Different subsets

9P3

987504 arrangements

10.3 – Using Permutations and CombinationsGuidelines on Which Method to Use

11.1 – Probability – Basic ConceptsProbability

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 ConceptsExample:If a single fair coin is tossed, what is the probability that it will land heads up?Sample Space:Event of Interest:P(heads) = P(E) = The probability obtained is theoretical as no coin was actually flipped

Theoretical Probability:

P(E) = number of favorable outcomes

S = {h, t}E = {h}1/2

total number of outcomesn(E)n(S)

=

11.1 – Probability – Basic ConceptsExample: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?

P(top) =

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

Empirical or Experimental Probability:

P(E)number of times event E occurs

number of times the experiment was performed

number of top outcomestotal number of flips

10100

= 110

=

11.1 – Probability – Basic ConceptsExample: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?

P(straight flush) =

This probability is theoretical as no cards were dealt.

number of possible straight flushestotal number of five-card hands

362,598,960

= 0.0000139 =

11.1 – Probability – Basic ConceptsExample: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?

P(female) =

This probability is theoretical as no experiment was performed.

number of possible female studentstotal number of students

835820 + 835

=

0.505

= 8351655

= 167331

P(female) =

11.1 – Probability – Basic ConceptsThe Law of Large NumbersAs 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 coinSpinnerRolling a die

11.1 – Probability – Basic ConceptsOdds

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).

Example:A to B A : B

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 ConceptsOdds

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

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

B to A B : A

5 : 1 What is the probability against rolling a 2 on a fair six-sided die?

5/6

11.1 – Probability – Basic ConceptsOdds

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?

Example:

200 – 10 = 10 : 190

What is the probability of winning the television?10/200

190 Unfavorable outcomes

10 Favorable outcomes

= 1 : 19

1/20= = 0.05

11.1 – Probability – Basic ConceptsConverting Probability to Odds

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

Example:

100 – 43 =

43 : 57 The odds for rain today:

57Unfavorable outcomes:

P(rain) = 0.43Of the 100 total outcomes, 43 are favorable for rain.

57 : 43 The odds against rain today:

11.1 – Probability – Basic ConceptsConverting Odds to Probability

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

Example:

16 : 9 The odds for completing the course:

P(completing the course) =

Favorable outcomes + unfavorable outcomes = total outcomes 16 + 9 = 25

= 0.64