**Ch 12 - Statistics**

**MATH017**

**Finite Mathematics**

**Lessons**

**Ch 2 Sets**

Ch 3 Logic

Ch 10 Counting Techniques

Ch 11 Probability

Ch 12 Statistics

Ch 3 Logic

Ch 10 Counting Techniques

Ch 11 Probability

Ch 12 Statistics

**An introduction to sets, logic, and probability.**

**Ch 10 - Counting Techniques**

**Ch 11 - Probability**

**Ch 2 - Sets**

**Ch 3 - Logic**

**Syllabus**

Instructor: Abbey Auxter, MS (aea15@psu.edu)

Phone: (215) 881 - 7546

Office: Sutherland 324 TR 12:20 – 1:30 pm, and by appt.

Class Time: TR 9:30 – 10:45 am

Class Room: Sutherland 222

**Tech World**

**Grades**

**Assignments**

**Exams**

Homework

Not graded.... but will still reflect in your grade

Quizzes

5 of them; Online

Middle of each chapter

3 tries

Online Assignments

2 throughout semester

Completion AND effort

Exams

5 (1 per chapter)

Final

Cummulative

Make-up

Emergencies only!

Notify me

Testing Center

Before next class

Exam Rules and Regulations

Closed Note

Calculator

Cannot leave

No talking, looking, cell phones

PSU E-mail

Use it

Canvas

Create acct

Quizzes

Online Assignment

Announcements

Grades

• Quiz Total = 50 pts

• Online Assignment = 25 pts

• Exam 1 = 65 pts

• Exam 2 = 65 pts

• Exam 3 = 65 pts

• Exam 4 = 65 pts

• Exam 5 = 65 pts

• Final Exam = 100 pts

TOTAL = 500 pts

February

March

May

April

February

January

Chapter 10 - Counting Techniques

10.1 The Sequential Counting Principle

10.2 Permutations

10.3 Combinations

10.4 Miscellaneous Counting Methods

Chapter 11 - Probability

11.1 Sample Spaces and Probability

11.2 Counting Techniques and Probability

11.3 Computation of Probabilities

11.4 Conditional Probabilities

11.5 Independent Events

11.6 Odds and Mathematical Expects

Chapter 12 - Statistics

12.1 Sampling and Frequency Distributions

12.2 Measures of Central Tendency

12.3 Measures of Dispersion

12.4 The Normal Distribution: A Problem Solving Tool

Final Exam

EVERYTHING!!!

Chapter 2 - Sets

2.1 Sets: A Problem Solving Tool

2.2 Set Operations

2.3 Venn Diagrams

2.4 The Number of Elements in a Set: A Problem-Solving Tool

2.5 Infinite Sets

Chapter 3 - Logic

3.1 Statements

3.2 Truth Tables: A Problem Solving Tool

3.3 The Conditional and the Biconditional

3.4 Variations of the Conditional

3.6 Truth Tables and Validity of Arguments

Course Schedule

2.1 Sets: A Problem Solving Tool

What is a set?

A set is a

well-defined

collection of objects, called

elements

or members

A = {4 mths, 2 yrs, 3 yrs, 4 yrs, 9 yrs}

"A is the set of the ages of Professor Auxter's nieces and nephews"

Uppercase letters denote a set

(lowercase letters denote an element)

What do we mean by "well-defined"?

Clear whether an object is an element of a particular set

For example, which of the following sets are well-defined?

Multiples of 2

Interesting Professors at PSU Abington

Good classes at PSU Abington

Current students in M017 at PSU Abington

Numbers that can be substituted for x so that x + 4 = 5

denotes that an object is an element of a particular set

denotes that an object is not an element of a particular set

How do we describe a set?

Describing sets in 3 ways

i. Verbal or written description

ii. Listing the elements of the set within braces (roster method)

iii. Set-builder notation

What if there are no elements in the set?

i.e. {x | x is a counting number between 8 and 13, and x is divisible by 7}

No number fulfills that, so we call it an empty set denoted as {} or

Warning: { } is not a thing!

(Can also be called the null set)

Why 3 ways? Why not just one?

Because not every set can be written just one way...

i.e. some well known sets we will revisit

Can sets be = ?

Two sets A & B are equal, denoted A = B, if they have the same elements (not necessarily written in the same order)

Determine whether the following sets are equal

i. {1, 2, 3} and {3, 1, 2}

ii. The set of digits in the year in which the Declaration of Independence was adopted and the set {1, 7, 7, 6}

iii. {2, 4, 6, 8} and the set of even counting numbers

Sets in sets

There is something called a Subset. A is a subset of B (denoted A B) if every element of A is also an element of B.

i.e. A = {y, z}, B = {x, y, z}, and C = {z}.

Which of the following (to the right) are true?

Also stated, the set A is a subset of B if there is no element of A that is not an element of B

This tells us that the empty set is a subset of every set.

Is the set {A, B, B, E, Y} a subset of the letters in my first name?

Is it a proper subset?

A is said to be a proper subset of B, denoted A B, is a subset of B but A B (A is not equal to B)

i.e. C = {chocolate, vanilla, strawberry}, therefore, {chocolate}, {vanilla}, {strawberry}, and {} are all elements of C and thus there are proper subsets of C. What about: {chocolate, strawberry}, {vanilla, strawberry}, and {chocolate, vanilla, strawberry}?

The Universal set, U - the set of what is being discussed.

i. i.e. If we are discussing integers, then U = {…, -2, -1, 0, 1, 2, …}

ii. i.e. If we are discussing students in this class right now, then U = {name, name, name, …}

Memory device: contained in...

the line can be thought of as

equal (like in less than or equal)

Number of subsets

Knowing what we do… how many subsets are there for the set U = {a, b, c}?

(Don’t forget the null set)

So there are 3 elements in this universal set

How might that relate to the number of subsets?

Maybe one set wasn’t enough to determine this

How many subsets are there for the set B = {2, 4, 6, 8}?

How many subsets are there for the set C = {24, 26, 28, 29, 31}?

It may not be super obvious but all of these follow a rule

The number of subsets of a set with n elements is 2^n

Therefore, is a set has 8 elements, how many subsets doe that set have?

Where does this come into play?

Problem-Solving: You are making cake pops. You have three icing colors to decorate the pops with C = {red, white, blue}. You don’t want any of the cake pops to look the same. How many different cake pops can be made?

How to approach ALL application problems...

R

ead the problem

S

elect the unknown

T

hink of a plan. What is the given? What do you do?

U

se your knowledge to carry out the plan

V

erify the answer

2.2 Set Operations

Intersection vs. Union

What defines the intersection of two streets? What about two groups?

For example, I am both a teacher and a student.

If A = {x | x is a Ph.D. student at Temple University in 2014} &

B = {x | x is a PSU Math teacher in 2014}, which set do I belong to?

The intersection of two sets - If A and B are sets, the

intersection

of A and B, denoted by A B, (read "A intersection B"), is the set of all elements that are common to both A and B.

U

A B

A

B

What defines a union? Think of a wedding... the union of two entire people... or families.

For example, the marriage of a Jackson and Robinson.

If J = {x | x is a member of the Jackson family} &

R = {x | x is a member of the Robinson family}. If they are entered into a union (or united), who is a member of that union?

The union of two sets - If A and B are sets, the

union

of A and B, denoted by A B, (read "A union B"), is the set of all elements that are either in A or in B or in both A and B.

U

A

B

If A = {t, g, i, f} and B = {f, y, i}, find the intersection and find the union of A and B.

Complement

Not the same as compliment :(

Let U be the universal set, and let A be a subset of U. The complement of A, denoted A' (read "A prime" or "A complement"), is the set of elements in U that are not in A. That is to say,

This set (the complement of A) can also be shown as U - A

U

A

B

Let U = {1, 2, 3, 4, 5, 6}, A = {1, 3, 5}, and B = {2, 4}

i. A’

ii. B’

iii. A’ B’

iv. A B

What if we try something CRAZY.... like (A B)' ?

U

A B

A

B

This is called De Morgan's Law

For any sets A and B, (A B)' = A' B' and (A B)' = A' B'

Let U = {a, b, c, d, e, f},

A = {a, c, e}, B = {b, e}, and C = {a, b, d}

Find (A B) C'

Find (A C)'

What happened to normal math symbols?

The minus sign lives!!!!

Difference of sets

If A and B are two sets, the

difference

of A and B, denoted A - B, is the set of all elements that are in A and not in B.

U

A

B

Idk if you noticed but before we said U - A = A'....

Here we have A - B = A B'

There are a couple ways of stating or understanding these

Give this a try...

Let U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3, 4}, and B = {1, 2, 5}.

Find...

U - A

A'

A - B

B - A

How we can use all this mumbo jumbo!

Clarifying weird situations

In 1900, Karl Landsteiner identified four blood groups as A, B, AB, and O (which is neither A nor B). The figure below shows how blood types are passed from parents to a child.

If one parent has type A blood and the other has type O blood, which blood types could their child have?

If a child has type B blood, what are the only possible blood types of the parents?

What combinations are missing from the diagram?

What blood type in a child would make it the most difficult to identify the paternity?

2.3 Venn Diagrams

Being sure of equality

How can we be sure that ?

PROVE IT! (Using our new friends Venn diagrams :)

Verify De Morgan's Law Verify distributive law

R

ead

S

elect unknown

T

hink of plan

U

se techniques

V

erify

Why a Venn diagram?

o Cuz math is confusing!!

o Venn diagrams/Euler circles/Euler diagrams can help visualize sets

o Can demonstrate rules

How to use this awesome tool!

The top 10 most popular social network websites according to EBizMBA (set E) and by the reviews they got (set R) are as follows (in ranking order 1-10):

E = {Facebook, Twitter, Linkedin, Ning, Tagged, Classmater, hi5, Myyearbook, Meetup}

R = {Facebook, Myspace, Bebo, Friendster, hi5, Orkut, Perfspot, Zorpia, Netlog, Habbo}

Write the set E intersect R

Which website would be the best to advertise on?

How a Venn diagram?

1. Rectangle = universe

2. Label circles (& insert elements into proper places, if necessary)

3. Shade to demonstrate what you are looking for

For example, from the diagram shown which region(s) represent(s)…

• A

• A’

• A intersect B

• A union B

• A’ intersect B’

• A' intersect B

• A – B

U

1

A

B

3

2

4

2.4 The Number of Elements in a Set:

A Problem Solving Tool

2.5 Infinite Sets

Cardinal number

If A = {1, 2, 3} and B = {2, 4, 6}, then n(A) = 3 and n(B) = 3 and since 3 = 3, n(A) = n(B) (the cardinal numbers are equivalent)

The cardinal numbers are the same (the sets are not equal)

How can we determine if the cardinal numbers are the same?

Venn diagrams -> # of Elmnts

Venn diagrams -> Life

If A is a set the number of elements contained in that set, commonly known as the cardinal number, is denoted n(A).

For example, if A = {m, a, t, h}, then n(A) = 4.

Revisiting number of elements

U

After looking at this we can deduce that...

Let's try this... In Germantown Academy 900 students are registered for the SAT, 600 are registered for the ACT, and 200 are registered for both.

What is the total number of registered students?

How many students are taking the SAT only?

How many students are taking the ACT only?

(Hint: RSTUV)

U

Another... There are 75 students taking MATH017 and 62 students taking BIO114. 46 of the MATH017 students are not taking BIO114.

What is the total number of students in the two classes?

How many are taking MATH017 only?

How many are taking BIO114 only?

Environmentalism... Yay!

To estimate the number of persons interested in recycling aluminum cans, glass, and newspapers, a company conducts a survey of 1000 people and finds that...

200 recycle glass (G) 300 recycle paper (P)

450 recycle cans (C) 50 recycle cans and glass

15 recycle paper and glass 60 recycle cans and paper

10 recycle all three

How many people do not recycle at all?

How many people recycle cans only?

U

G

P

C

In a survey of 100 students, the number taking Algebra (A), English (E), and Philosophy (P) are shown below:

How many students are taking...

Algebra or English but not both?

Algebra or English but not Philosophy?

one or two of these courses but not all three?

at least two of these courses?

at least one of the courses?

U

E

P

A

30

26

6

10

8

7

5

8

i. Let C = {Franklin and Marshall College, Drexel University, Temple University}. True or False?

1. Drexel University C

2. D C

3. Temple University C

4. Franklin and Marshall College {Franklin and Marshall College, Drexel University, Temple University}

5. C C

Equivalence

What is a one-to-one correspondence, you ask?

For example,

a e i o u

demonstrates a one-to-one correspondence

1 2 3 4 5

so {a, e, i, o, u} ~ {1, 2, 3, 4, 5}

N was weird...

N was an infinite set... A set is

infinite

if it is the equivalent to one of its proper subsets

N ~ E (E was a proper subset in that it didn't have the same elements)

A man named Georg Cantor came up with the math discipline, set theory. He assigned the transfinite cardinal ("aleph null", 1st letter of Hebrew alphabet) to infinite sets. (i.e. finite sets have cardinal numbers, infinite sets have the transfinite cardinal number)

Show that S = {1, 4, 9, 16, 25, ..., n^2) is equivalent to N.

Find the cardinality of S.

Show S is infinite.

What about a longer set? Let's start at the beginning...

N = {1, 2, 3, ...}

E = {2, 4, 6, ...}

This is a demonstration of one-to-one correspondence... one element for one element

But how many elements are in N? Do we really have to do this for every one? Wouldn't you think that N would have twice as many elements?

If two sets A & B can be placed into a one-to-one correspondence with each other, the two sets are said to be

equivalent

, denoted A

~

B

Show that A = {!, @, #, $} and B = {%, &, *, +} have the same cardinal number

N = {1, 2, 3, 4, ... , n, n + 1}

E = {2, 4, 6, 8, ... , 2n, 2(n + 1)}

For every element in N there is a corresponding element in E... i.e. one-to-one correspondence

N ~ E

In mathematics, when a set is equivalent to N (the set of counting numbers), the set is said to be

denumerable

3.1 Statements

3.2 Truth Tables:

A Problem-Solving Tool

3.3 The Conditional & the Biconditional

3.4 Variations of the Conditional & Implications

3.6 Truth Tables & Validity of Arguments

What is a statement?

a declarative sentence that can be classified as true or false (but not both simultaneously)

A statement is not a question (asked), command (given), or exclamation (shouted)!

Statements

If you build it, they will come

Either you study daily or you will get an F in the class

Boston is the capital of Massachusetts

Nonstatements

What time is it?

Elvis for President

This statement is false

(the above is a paradox - a

selfcontradictory sentence)

These are true/false statements

Some are easily checked, others not so much

Cooky Statements!

Simple statement - says one thing

Ex 1. Leslie has a dog (d)

Ex 2. Leslie has a cat (c)

Compound statement - says more than one thing

Leslie has a dog

and

a cat

Leslie has a dog

or

a cat

And & Or are connectives denoted:

And =

Or =

How could we represent the above two compound statements using the letters and new symbols?

Conjunction - If two statements are combined by the word

and

(or an equivalent word such as

but

), the resulting compound statement is called a conjugate (denoted p q)

p: It is 10 degrees today

q: The heat is blasting

It is 10 degrees today and the heat is blasting.

Disjunction - If two statements are combined by the word

or

(or an equivalent word such as

otherwise

), the resulting compound statement is called a disjunction (denoted p q)

p: It is 10 degrees today

q: The heat is blasting

It is 10 degrees today or the heat is blasting.

Negation - When a given statement is true, its negation is false. When a given statement is false, its negation is true. (denoted ~p) [~ is called tilde]

p: It is 10 degrees today

It is not the case that it is 10 degrees today.

Rewrite as symbols...

Taylor is buying an iPad and Sam is buying a surface.

Cora is good at soccer but she is best at basketball.

We study for math or we fail the course.

Symbol <-> English

p: The sky is blue.

q: It is raining.

~ (p q) ->

~ p ~ q ->

~ q p ->

De Morgan's Laws are back!!!

Negation of Conjunction

~ (p q) = ~ p ~ q

Negation of Disjunction

~ (p q) = ~ p ~ q

p: Elvis is alive. & q: Elvis has left the building

Write the following in symbols

a) Elvis is alive and has left the building

b) Either Elvis is alive or he has left the building

c) Elvis is neither alive nor has he left the building

d) It is not the case that Elvis is alive and has left the building

Write as symbols...

a) You are a full-time student (f) or over 21 (o), and a resident of PA (r).

b) You are a full-time student, or over 21 and a resident of PA.

p: Demi likes Ashton. & q: Ashton likes Demi.

Write the following...

1) ~ (p q)

2) ~ p ~ q

3) ~ p ~ q

Negation w/ all, none, some

Universal quantifiers - all, none, every

Existential quantifiers - some, there exists at least one

WHAT DOES THIS MEAN?!?!?!

Existential (of or relating to existence)

Quantifier (an expression (e.g., all, some ) that indicates the scope of a term to which it is attached)

STATEMENT

All a's are b's

No a's are b's

NEGATION

Some a's are not b's

Some a's are b's

Negate the following...

m: Everyone likes math

~ m:

p: No one goes to PSU

~ p:

g: Some of us will graduate

~ g:

Seriously? What is this for?

Suppose you apply for a credit card and must sign something that says... "I request that a Visa account be opened (o) and cards be issued as indicated (i), and I authorize the bank to receive (r) and exchange (e) information and investigate (n) the references and data (d) collected pertinent to my creditworthiness".... Huh? What if you have to sign this? What doe it say?!?!

Try writing this using symbols

Here is a tip from a software manual.

Some of the things you can use the Control Panel for are changing your screen colors, installing or changing settings for hardware and software, and setting up or changing settings for a network.

Indicate in symbols how you can use the Control Panel. (Hint: Determine how many statements are in the original)

A conjunction & its truth value

A disjunction & its truth value

A negation & its truth value

Truth table construction

Equivalence by truth table

Who here has done their taxes?

Have you ever really read the instructions for your 1040 tax form?

To determine whether a person qualifies as a dependent five qualifications must be met including:

Here we were able to translate to a disjunction and, recall, what makes a disjunction true is if any of its components is true. (ie. any of the parentheses).

Suppose you are offered a position in a firm requiring that:

p: An applicant must be at least 18 years of age.

AND

q: An applicant must be a college graduate.

Both requirements must be fulfilled in order to be eligible/for the conjunction to be true.

We can create 4 possible combinations with p and q

p true, q false

p true, q true

p false, q true

p false, q false

Which one(s) of these combinations will yield a true p q?

Note that other English words such as but, nevertheless, still, however, and so on are sometimes used in place of the connective and.

Let p be “I will pass this course.”

Let q be “I will flunk this course.”

Form the disjunction of p and q.

Can you do both?

What does this tell us about the truth table?

Consider the two statements:

m: I will study Monday.

s: I will study Saturday.

Form the disjunction of m and s.

Can you do both?

What does this tell us about the truth table?

If we compare the disjunctions, it is clear that in the first disjunction, only one of the two possibilities can occur: I will either pass the course or I will not pass the course.

However, in the second disjunction,, I have the possibility of studying Monday or Saturday or both. The meaning of the second usage is clarified by replacing the word or by and/or.

Instead of arguing which usage should be called the disjunction of the two statements, we shall refer to the or used in Example 1 as the exclusive or or the exclusive disjunction.

The or used in the second disjunction (study Monday or Sunday) will be called the inclusive or or the inclusive disjunction and will be denoted by V.

Let p be the statement “I will go to college.” Express the statement ~p in words.

If p is true, what does that say about ~p?

If p is false, what does that say about ~p?

When filing your Form 1040A, you will be classified as single if any one of the following is true:

You were not married (~m).

You were legally separated under a decree of divorce (d) or of separate maintenance (s).

You were widowed before January 1 (w) and did not remarry (~r).

a) Write in symbols the conditions under which you will be classified as single.

b) " " not single.

Construct the truth table for the statement ~(p ~q).

How about ~ p q

What does equivalence mean? --> The condition of being equal in value, worth, function, etc.

SO how does this transfer to statements? And how do we prove it using truth tables?

(a) Show that (~ p q) r <=> (~ p r) (q r).

(b) Let p = “You recycle paper,” q = “You recycle glass,” and r = “You are an environmentalist.”

If we know that you recycle paper, under what condition(s) will the statement (~ p q) r be true?

Are the statements equivalent?!?!?

Equivalence continued

Show that the two statements are equivalent... ~ (p q) and ~ p ~ q

Conditional in symbolic form & truth value.

Biconditional in symbolic form & truth value.

Equivalency & Negation

of Conditionals

Conditional? "If p, then q" (denoted p -> q)

p is considered the antecedent & q the consequent

Write the statement in the picture in symbolic form:

v: the pump runs very slowly r: release nozzle trigger

c: count to 20 t: try again

When is the above conditional false?

What does "Bi" mean?

Here we are using the conditional twice in the following form:

"If Prof Auxter is awake (a), then she is thinking about math (m) and if Prof Auxter is thinking about math, then she is awake"

OR

: "Professor Auxter is awake if and only if she is thinking about math." Or vice versa

Biconditionals

are denoted p <-> q

How would the above biconditional be represented?

Biconditional is true when and only when p and q have the same truth values (i.e. both true or both false)

Conditional & Biconditional Practice

Truth values of Conditional

Give the truth value of the following...

If Tuesday is the last day of the week, then the next day is Sunday.

If Tuesday is the third day of the week, then Wednesday is the fourth day of the week.

Is the statement (3 + 5 = 35) <-> (2 + 7 = 10) true or false?

Let p be "x is a fruit," and let q be "x is ripe." What makes the statement p -> q false?

Since p is false, p -> q is true.

(The statement is only false when p is True and q is False)

Since p & q are both true, p -> q is true.

(The statement is legit... cause and consequence)

This is very similar to what we did with proving using

Venn diagrams.... We can prove equivalence using truth tables.

Show that the statements p -> q and ~ p q are equivalent; that is, show that (p -> q) <=> (~p q)

Before we were looking for identical Venn diagrams... now we want identical truth values.

This new information is a breakthrough because it allows us to understand he negation of a conditional in terms we are already familiar with.

Equivalency & Negation of Conditionals Practice

Equivalency

Write the equivalent statement as a disjunction and prove it is equivalent using truth tables.

If you work, you have to pay taxes.

Converse, inverse, & contrapositive of a conditional statement

Conditional Equivalents

Tautology & contradiction

Implications

Unlike the conjunction and disjunction, a conditional is not commutative... Remember p and q is the same as q and p.

If p, then q is NOT the same as If q, then p.

But we do have special names for these types of situations.

Before we noticed that some statements can be equivalent if their truth values were the same....

Let t be “I will tip” and let s be “Service is good.” Write in symbols and in words:

a. The conditional “If s, then t” c. The inverse of s -> t

b. The converse of s -> t d. The contrapositive of s -> t

The words necessary and sufficient are often used in conditional statements.

To say that "p is sufficient for q" means that when p happens (is true), q will also happen (will also be true). Hence, “p is sufficient for q” is equivalent to “If p, then q.”

Similarly, the sentence “q is necessary for p” means that if q does not happen, neither will p. That is, ~q -> ~p.

The statement ~q -> ~p is equivalent to p -> q, so the sentence “q is necessary for p” is equivalent to “If p, then q.”

p is necessary and sufficient for q

q is necessary and sufficient for p

q if and only if p

are all equivalent to the statement “p if and only if q” and can be symbolized by p <-> q.

Let s be “You study regularly” and let p be “You pass this course.”

Translate the following statements into symbolic form.

a. You pass this course only if you study regularly.

b. Studying regularly is a sufficient condition for passing this course.

c. To pass this course, it is necessary that you study regularly.

d. Studying regularly is a necessary and sufficient condition for passing this course.

e. You do not pass this course unless you study regularly. (Hint: a unless b means ~b -> a.)

a. p -> s

b. Because s, studying regularly, is the sufficient condition, write s -> p.

c. Since s is the necessary condition, write p -> s.

d. p <-> s or s <-> p.

e. “You do not pass this course unless you study regularly” can be written as ~p unless s, which means ~s -> ~p.

Show the following by means of a truth table:

a. The statement p v ~p is a tautology.

b. The statement p ^ ~p is a contradiction.

Another relationship between statements that is used a great deal by logicians and mathematicians is that of implication.

Show that [(p -> q) ^ p] => q.

Solution:

First method

By the definition of implication we must show that [(p -> q) ^ p] -> q is a tautology.

A conditional is true whenever the antecedent (usually p but in this case [(p -> q) ^ p] ) is false, so we need to check only the cases in which the antecedent is true.

Thus, if (p -> q) ^ p is true, then p -> q is true and p is true. But if p is true, then q is also true (why?), so both sides of the conditional are true.

This shows that the conditional is a tautology, and thus, (p -> q) ^ p implies q.

Second method

A different procedure, which some people prefer, uses truth tables to show an implication. In order to show that a => b, we need to show that a -> b is a tautology.

In our case, we have to show that [(p -> q) ^ p] -> q is a tautology. We do this by constructing a truth table where column 1 is p -> q, which is false only when p is true and q if false (row 2).

Column 2 corresponds to the conjunction (p -> q) ^ p, which is true only when both p -> q and p are true (row 1).

In column 3, we simply copy the truth values of q and finally, in column 4, we look at the conditional .

Write an argument in symbolic form.

Use truth tables to determine validity

Supply a valid conclusion to a given argument using all premises.

An argument is

Valid

if the conclusion is true whenever all the premises are assumed to be true.

If an argument is not valid, it is said to be

invalid

.

To help us better create a truth table to show validity we write the argument in symbolic form.

Symbolize the following argument:

Solution:

Let e be “A whole number is even” and let o be “A whole number is odd.”

Then the argument is symbolized as

Now let's try this one:

If today is Tuesday, then I will go to math class.

Today is Tuesday.

I will go to math class

Solution:

1.

2.

Let e be “A whole number is even” and let o be “A whole number is odd.”

3.

Use a truth table to determine the validity:

We first write the argument in symbolic form.

“All dictionaries are useful books” is translated as “If the book is a dictionary (d), then it is a useful book (u).”

“All useful books are valuable” means “If the book is a useful book (u), then it is valuable (v).”

“All dictionaries are valuable” is translated as “If the book is a dictionary (d), then it is valuable (v).”

The nice thing about this example is we know that an argument of the above form is always true so when we get long arguments like.....

We can use that knowledge

In using a truth table to check the validity of an argument, we need to examine only those rows where the premises are all true.

Suppose you know the following to be true:

1. If Alice watches TV, then Ben watches TV.

2. Carol watches TV if and only if Ben watches.

3. Don never watches TV if Carol is watching.

4. Don always watches TV if Ed is watching.

Show that Alice never watches TV if Ed is watching.

Solution:

Let a be “Alice watches TV.”

Let b be “Ben watches TV.”

Let c be “Carol watches TV.”

Let d be “Don watches TV.”

Let e be “Ed watches TV.”

Write the preceding arguments in symbols:

1. a -> b

2. c <-> b or b <-> c

3. c -> ~d

4. e -> d, or equivalently, ~d -> ~e

By arranging these in an optimal order we can see

1. a -> b

2. b <-> c

3. c -> ~d

4. ~d - > ~e

And knowing what we learned before about

arguments of this type.... a -> ~e.

10.1 The Sequential Counting Principle (SCP):

A Problem-Solving Tool

ALL THE BREAKFASTS!!

But how many is that?!?

Suppose we go out to breakfast... the server asks, “How do you want your eggs? Fried ( f ), poached (p), or scrambled (s)? Rye (r) or white toast (w)? Juice (j) or coffee (c)?” How many choices do you have?

We can see that this is the same as saying

# of Egg options x # of toast options x # juice options

3 x 2 x 2

= 12 possible breakfast combinations

Suppose we had a 2 question Socrative and both questions were True or False types. Using a Tree Diagram, show how many possible combinations of answers could be given.

Say you are taking English, Math, Biology, Art, and Philosophy. Draw a tree diagram listing your first and second choices (all possibilities).

SCP

We already figured this out utilizing the Tree Diagrams...

Johnny’s Homestyle Restaurant has 12 different meals and 5 different desserts on the menu. How many meal choices followed by a dessert choice does a customer have?

Suppose we added the choice of coffee, water, and hot chocolate to go with the dessert?

Three dice are thrown.

How many different outcomes are possible?

If you picked 4 as your lucky number, in how many ways could you get a sum of 4?

What is this for?!?

The techniques we have studied can be used to determine the cost effectiveness of alternative courses of action.

Suppose your doctor tells you that you can be treated with drug A or B.

You may then need a second visit (or not) depending on your tolerance for the drug. The costs for drugs A and B are $80 and $50, respectively. Your doctor charges $100 per visit.

(a) How many choices are possible?

(b) What are the highest and lowest possible costs for your treatment?

**10.2 Permutations**

Permutations

An ordered arrangement of r objects selected from n objects without repetition.

Denoted P (n, r); n is available & r is chosen

or nPr (read permutations of n objects selected r at a time

A lab wants to experiment with animals by giving them 3 different medicines chosen from a group of 5 medicines and studying the results. How many lab animals are needed to perform the experiment?

From what we saw previously we know we can find the answer by

5 x 4 x 3 = ?

Is giving on animal Medicines A, B, then C the same as giving A, C, then B?

In this section we will learn a more efficient form of computing Permutations called

Factorial Notation.

Compute the following:

a. P(6, 6)

b. P(7, 3)

c. P(6, 2)

r = # of blanks/choices

Use the factorial formula for P(n, r) to compute: P(7, 3)

Complementary Counting Principle

It is sometimes easier to find n(A) by n(U) - n(A'). This gives us the complementary counting principle.

How many ways are there to select at least 1 male puppy if 4 puppies are available?

The only alternative to selecting at least 1 male puppy is selecting no male puppies; that is, the 4 puppies are all females.

This is just one of all the possible cases.

There are 4 places to fill, with 2 possible choices for each place (male or female).

n(U) = 2 x 2 x 2 x 2 = 16 possible puppy combinations. Only one of those has them all female so 16 - 1 = selecting at least one male puppy.

Additive Counting Principle

Another useful counting principle is the additive counting principle, giving the number of elements in the union of two sets.

How many 2-digit counting numbers are divisible by 2 or by 5?

Let A be the set of 2-digit numbers divisible by 2, and let B be the set of 2-digit numbers divisible by 5.

We are looking for the number of digits that are divisible by 2 or 5, that is n (A U B),

Using the SCP, for 2-digit numbers divisible by 2, the first digit can be any digit from 1 to 9 (9 choices), and the second digit can be 0, 2, 4, 6, or 8 (5 choices).

Since is the set of numbers divisible by both 2 and 5, the first digit can still be any digit from 1 to 9 (9 choices), and the second digit can only be 0 (1 choice).

And we use all of this for...

Suppose Boston Market has 16 side dishes.

a. In how many ways can you select 3 different dishes?

b. How many permutations of 16 objects taken 3 at a time are there?

c. Suppose you select carrots, potatoes, and broccoli. If you had selected broccoli, potatoes, and carrots, would the end result be different?

d. Is the order in which you select your side dishes important?

10.3 Combinations

A Permutation is an ordered arrangement of n distinguishable objects, taken r at a time and with no repetitions.... A combination differs in one way.... it is not ordered.

Here are 4 incentives to buy a Smartphone:

(a) Decreasing price

(b) Increased job productivity

(c) Help you stay organized

(d) Best platform for social networking

Consider the set of four incentives S = {a, b, c, d}.

(a) How many combinations of 2 incentives are possible using elements of the set S?

(b) How many permutations of 2 incentives are possible using elements of the set S?

(c) Which will give you more incentives, the permutations of 2 incentives or the combinations of 2 incentives?

(d) How many subsets of 2 elements does the set S have?

How many subsets of at least 3 elements can be formed from a set of 4 elements?

How many subsets of at least 4 elements can be formed from a set of 4 elements?

Suppose you are asked to find the number of combinations of 10 objects, 8 at a time.

What if you were asked to find the number of combinations of 10 objects, 2 at a time?

Suppose we are selecting 3 side dishes from 16 available dishes and count the number of choices we have for our 3 side dishes

(a) without any repetitions (we have to pick 3 different dishes).

(b) with one repetition (say aba or ccd or eff).

(c) with 3 repetitions (say aaa, bbb, or ccc).

10.4 Miscellaneous Counting Methods

Let's start thinking...

“Counting” on Winning the Lottery

Suppose you play a lottery in which you pick 4 digits

and you win if the digits are drawn in the exact order you have chosen.

To find your chance of winning, you must find in how many ways you can select the 4 digits. The number of ways in which you can fill the 4 blanks.

____ ____ ____ ____

In many state lotteries you pick 6 different numbers from a set of 49.

You win the jackpot if you match (in any order) the 6 winning numbers.

How many chances of winning do you have now?

Combinations & Permutations

Do you want to save the environment? See the following tips you can use.

Tips for the Office

1. Copy and print on both sides of paper

2. Reuse items (envelopes, paper clips)

3. Use e-mail instead of regular mail

4. Use ceramic mugs instead of paper cups

Tips to Save Water

5. Check and fix water leaks

6. Use water-saving devices on faucets

7. Install low-flow shower head

8. Wash and dry only full loads

9. Replace toilets with efficient ones

(a) How many

priority

lists can you make if you pick one tip from each list?

(b) How many

different

lists can you make if you pick 2 Tips for the Office and

then

2 Tips to Save Water?

(c) In how many ways can you

prioritize

(order) the 4 items in the Tips for the Office?

(d) How many

sets

of tips can you make with the 5 items on the Tips to Save Water list?

(e) How many

sets

of tips can you make if you pick 3 tips from the second list?

Permutations of Nondistinct Objects

IS REPETITION

(OR REPLACEMENT) ALLOWED?

YES!

USE SCP

NO!

IS ORDER IMPORTANT?

YES!

nPr

NO!

nCr

How many

different

license plates can be made using 3 letters followed by 2 numbers (0-9)?

How many different arrangements can be made using all of the letters in the word MOVIE?

A basketball coach must choose 4 players to play in a particular game. (The team already

has a center) In how many ways can the remaining 4 positions be filled if the coach

has 8 players to pick from?

Out of 8 children, how many ways can a family have exactly one boy?

In how many ways can a committee consisting of 7 men and 5

women be selected from a group consisting of 14 men and

15 women?

What if I asked you, how many ways you could arrange the letters in A = {L, I, L, Y}?

There are 2 L's.... are they counted as the same or different?

Let's look at if they are considered different...

LI

L

Y

L

ILY I

L

LY Y

L

LI LILY LILY

ILLY

YLLI

LIY

L

L

IYL I

L

YL Y

L

IL

LIYL LIYL

ILYL

YLIL

L

L

IY

L

LIY IL

L

Y YL

L

I

LLIY LLIY

ILLY

YLLI

L

L

YI

L

LYI ILY

L

YLI

L

LLYI LLYI

ILYL

YLIL

LY

L

I

L

YLI IY

L

L YI

L

L

LYLI LYLI

IYLL

YILL

LYI

L

L

YIL IYL

L

YIL

L

LYIL LYIL

IYLL

YILL

24 different arrangements 12 different arrangements

Much like with Combinations we need to divide by a factorial... but what factorial this time?

In this case (they are all caps otherwise that would be a distinction) we have 2 L, 1 I, 1 Y so 4!/2!1!1! = 24/2

Suppose you were trying to find how many ways yo could write the last name in the San Fransisco phonebook: Zzzzzzzzzra.

A lawyer handling legal cases “first identifies the factual and legal uncertainties in a case and then decides: Should we litigate or settle?” If we litigate, we can lose or win a

summary judgment.

If we lose, the jury finds liability (high, medium, or low) or there may not be any liability.

Draw a tree diagram and show all the possibilities for the case.

Application of a Tree Diagram

11.1 Sample Spaces & Probability

11.2 Counting Techniques & Probability

11.3 Computation of Probability

11.4 Conditional Probabilities

11.5 Independent Events

11.6 Odds & Mathematical Expectation

What is 'Probability'?

Some probabilities you way be interested in...

EVENT

PROBABILITY

Struck by lightning 1/576,000

Audited by IRS 1/175

Getting the flu this year 1/10

Cancer (lifetime, men) 1/2

Cancer (lifetime, women) 1/3

What is the probability of winning the lottery given the parameters (choosing 6 numbers from 1-49 non-repeating and any order)? What if you didn't buy a ticket? What if you bought 2?

Probability is the mathematical estimate of the likelihood that a particular event will occur.

Other than trying to hit the jackpot... why do I care about probability?

Probability is used to determine whether a missile will hit it's target, determine insurance premiums, to make important business decisions, and in weather.

Theoretical Probability

Empirical Probability

A fair coin is tossed; find the probability of getting heads.

Our intuition tells us the following:

1. When a fair coin is tossed, it can turn up in one of 2 ways. Assuming that the coin will not stand on edge, heads and tails are the only 2 possible outcomes.

2. If the coin is balanced (and this is what we mean by saying “the coin is fair”), the 2 outcomes are considered equally likely.

3. The probability of obtaining heads when a fair coin is tossed, denoted by P(H), is 1 out of 2. That is, P(H) = 1/2 .

Suppose that a fair coin is tossed 3 times. Can we find the probability that 3 heads come up? As before, we proceed in three steps:

1. Find the total number of possible outcomes (here we can use a tree diagram

2. We determine that the 8 outcomes are equally likely (due to former knowledge)

3. The outcome of getting 3 heads P(HHH) is 1/8 (this is determined by # of favorable events/# of total possible outcomes

I want you to label 8 lines on your paper... 8 experiments. Each one will consist of you tossing your penny 3 times. If you get heads, tails, heads, record it on line 1 as HTH. Based on what we have calculated 1 (and only 1) of your 8 lines will have the outcome HHH

What happened with your experiments?

Did you get 1 HHH?

Does this mean probability is incorrect?

Theoretical Probability is based on theory and the idea that the outcomes are equally likely... Empirical Probability is known as experimental probability.... what actually occurs based on experiments/experience

Activity #2: I want you to toss the penny in the air 10 times. We know that theoretical probability will tell us to expect 5 tails and 5 heads. What is the empirical probability of getting tails? Why don't we always use this?

Application: An online survey of 324 people conducted by Insight Express asked the question, “What is your primary credit card?” The results are shown below.

If a person is selected at random from the 324 surveyed, what is the empirical probability that:

(a) the person’s primary card is MasterCard?

(b) the person’s primary card is Visa?

(c) Which event has the highest empirical probability? What is that probability?

(d) Which event has the lowest empirical probability? What is that probability?

(e) If you were the manager of a retail store and you can only accept two types of credit cards, which two cards would you accept?

Practice & Some other things

Let's go back to Activity #2... You tossed the penny 10 times. I asked you the empirical probability of getting tails. Without referencing the number of times you got not tails, how could you tell me the empirical probability of getting not tails?

Practice:

Ten balls numbered from 1 to 10 are placed in an urn. If 1 ball is selected at random, find the following probabilities:

(a) An even-numbered ball is selected (event E).

(b) Ball number 3 is chosen (event T).

(c) Ball number 3 is not chosen (event T' "T prime").

From this example we learn that (T union T') = U

and (T intersect T') = {}

Fun Fact about 'coin probability'

If you toss the coin more and more and compare the theoretical and empirical probabilities you notice something.

Law of Large Numbers

Tree diagram to find probability

Permutations and combinations to find probability

Applications involving probability

Have you heard of the witches of Wall Street? These are people who use astrology, tarot cards, or other supernatural means to predict whether a given stock will go up, go down, or stay unchanged.

Not being witches, we assume that a stock is equally likely to go up (U), go down (D), or stay unchanged (S). A broker selects two stocks at random from the New York Stock Exchange list.

(a) What is the probability that both stocks go up?

(b) What is the probability that both stocks go down?

(c) What is the probability that one stock goes up and one goes down?

Two cards are drawn without replacement from an ordinary deck of 52 cards. Find the probability that

(a) the cards are both aces.

(b) an ace and a king, in that order, are drawn.

Do you know the probability of getting four aces in a 5-card poker hand?

Since there are four aces in a deck, there is only 1 way of getting four aces if the fifth card can be any of the 48 remaining; hence, there are 1 C(48, 1) = 48 ways of getting four aces of the C(52, 5) = 2,598,960 possible poker hands.

Can you find the probability of getting three clubs and two diamonds?

An oil company is considering drilling an exploratory oil well. If the rocks under the drilling site are characterized by what geologists call a “dome” structure, the chances of finding oil are 60%. The well can be dry, a low producer, or a high producer of oil. The probabilities for these outcomes are given in the table.

(a) Draw a tree diagram for the data given in the table.

(b) What is the probability that the well is dry?

The diagram shows the number of states regulating smoking in various areas. If you select one of the 50 states at random, what is the probability that smoking is regulated in:

(a) Restaurants, P(R)

(b) Bars and Restaurants, P(B intersect R)

(c) Restaurants only, P(RO)

(d) Bars, P(B)

(e) Restaurants or Bars, P(R union B)

We have used the formula P(T

'

) = 1 - P(T) to calculate the probability of the complement of an event. We now see how this property is used in the field of life insurance.

Find the probability that a person who is alive at age 20 is

(a) still alive at age 70 if the person is a female.

(b) not alive at age 70 if the person is a female.

(c) still alive at age 70 if the person is a male.

(d) not alive at age 70 if the person is a male.

Example: Probability of rolling a 7 on a die labeled 1-6

Example: Probability of rolling a 5 on a die labeled 1-6 is 1/6

Example: The Venn Diagram coming up

Probability of two events that do NOT overlap.

A card is drawn from a deck of 52 playing cards. Find the probability that the card is either an ace or a red card.

An urn contains 5 red, 2 black, and 3 yellow balls. Find the probability that a ball selected at random from the urn will be red or yellow.

Why do you work? A Pew Research study concluded that the answer depends on your age. Forty-nine percent of persons in the 16-64 age group worked because they needed money (N), 20% because they wanted to work (W) and 31% because of both N and W.

(a) If a person between the ages of 16-64 is selected at random, find P (N U W)

(b) Are N and W mutually exclusive?

Consider buying a car! Dealerships have to do their own research on you, the consumer.

If you are the manager of a dealership, you want to know whether people who read the reports in Consumer Reports or Car and Driver are more likely to buy a car from you.

The first step is to conduct a survey of potential buyers.

Suppose the results of such a survey are as follows:

70% of the people read the report (R).

45% bought a car from you, the dealer (B).

20% neither read the report nor bought a car from you, the dealer.

You want to find the effect of reading the report (R) on buying a car from you (B). Thus, you must compare the probability that the person bought a car P(B) with the probability that the person bought a car given that the person read the report, this is denoted as P (B | R) and read as “the probability of B given R.”

It may be helpful to make a Venn diagram of the situation.

How do you find P(B | R)? We know we are basing our information out of the people that read the magazine and our favorable event is B. If someone bought a car but didn't read, is that considered a favorable event? So what goes where?

Assume that in Getwell Hospital, 70 of the patients have lung cancer (C), 60 of the patients smoke (S), and 50 have cancer and smoke.

If there are 100 patients in the hospital, and 1 is selected at random, then P(C) = 70/100 and P(S) = 60/100. But suppose a patient selected at random tells us that he or she smokes.

What is the probability that this patient has cancer?

In other words, what is the probability that a patient has cancer,

given that the patient smokes

?

The above red statement is what we call a restrictive condition. In this case it limits us to using S as the sample space.

What information do we need?

The number of smokers for the sample space

The number of smokers with lung cancer for the 'favorable' event.

Did we NEED a Venn diagram in this case?

Two dice were thrown, and a friend tells us that the numbers that came up were different. Find the probability that the sum of the two numbers was 4.

Solution:

Method 1

Let D be the event in which the two dice show different numbers, and let F be the event in which the sum is 4.

By equation (1),

Now, P (F intersect D) = 2/36 because there are two outcomes,

(3, 1) and (1, 3), in which the sum is 4 and the numbers are different, and there are 36 possible outcomes.

Furthermore,

So...

Method 2:

If you notice, this is the same as 2/30... number of favorable outcomes given the conditions/possible outcomes or sample space

Application: The Framingham Heart Disease Study focused on strokes and heart failure. The Table is based on this study and shows the number of adults (per 1000) aged 45– 74 with certain blood pressure types and the number of strokes in each category.

As we can see, the incidence of stroke for people aged 45–74 increases almost fourfold as blood pressure goes from normal to high (from 8 per 1000 to 31 per 1000).

Note: the numbers in the Table are all per 1000. This means that Table 11.5 gives approximate conditional probabilities.

The number 31 in the last line of Table 11.5 means that the probability that a person will have a stroke and has high blood pressure is about 31/1000

Find the probability that...

(a) a person in the 45–74 age group has a stroke (S), given that the person has normal blood pressure (N).

(b) a person in the 45–74 age group has a stroke (S), given that the person has borderline blood pressure (B).

(c) a person in the 45–74 age group has a stroke (S).

(d) a person has normal blood pressure (N), given that the person had a stroke (S).

Probabilities in Bingo:

The world’s biggest Bingo contest was held in Cherokee, North Carolina, and offered a $200,000 prize to any player who could fill a 24-number card by the 48th number called (there are 75 possible numbers in Bingo).

What is the probability that you would win this game? The probability that any given number on your 24-number card is drawn is 48/75, the probability of drawing a second number on your card is 47/74, and so on. To win, you must get all 24 numbers on your card in 48 draws.

The probability is

Note that the individual probabilities have been multiplied to find the final answer.

Here we will study

independent events

. If two events A and B are independent,

P(A intersect B) = P(A) * (B

The drawing of the numbers are not dependent on each other. Drawing one does not effect whether the next is more or less likely to happen (aside from the lessening of numbers available). Another example would be the rolling of 2 dice. If you roll a 6 with the first, it does not affect your chances of rolling a 6 with the second.

Have you been to a baseball game lately? Did anybody hit a home run? What are the chances of that?

In a recent year, the number of home runs in MLB was 5386 and the number of plate appearances 188,052, thus,

about 1 in 35 times, or more precisely, 0.02864. But what about the probability of four home runs in a row?

Assume that each players’ at bat is independent of each other and that the probability of a home run is P(HR) = 0.02864

(a) What is the probability that four home runs are hit consecutively?

(b) If instead of using 0.02864 as the probability of hitting a home run by each of the players, use the individual probability that each of the players hit a home run, 0.0608, 0.0403, 0.0369 and 0.0324, for Ramirez, Drew, Lowell and Varitek, respectively. What is the probability now that four home runs are hit consecutively?

(c) There have been about 170,000 MLB games since 1900 and only 5 times have four home runs been hit in a row. Based on this information, what is the probability, written as a fraction, that four home runs are hit in a row?

Which of the three above probabilities is right?

Stochastic Processes

A

stochastic process

is a sequence of experiments in which the outcome of each experiment depends on chance.

For example, the repeated tossing of a coin or of a die is a stochastic process. Tossing a coin and then rolling a die is also a stochastic process.

In the case of repeated tosses of a coin, we assume that on each toss there are two possible outcomes, each with probability .

In a recent year 16 named storms formed in the Atlantic. Of those, 5 became hurricanes and 3 (Don, Lee and Irene) hit the U.S. Based on this information, what is the probability

that a named storm:

(a) will become a major hurricane? (d) does not hit the U.S.?

(b) hits the U.S. in any single year? (e) does not hit the U.S. in the next 3 years?

(c) hits the U.S. in 3 consecutive years? (f) hits the U.S. at least once in the next 3 years?

Lottery Odds: Look at the information in Table 11.6. The odds of winning the first prize by picking 6 out of 6 numbers (there are 49 numbers to pick from) are said to be 1 in 13,983,816.

Odds do NOT = probability

Probability

of an event is a fraction whose numerator is the number of times the event can occur and whose denominator is the total number of possibilities in the sample space.

The

odds

in favor of an event are the ratio of favorable to unfavorable occurrences for the event.

Since the probability of winning the first prize is 1 to 13,983,816, the odds of winning the first prize are 1 to 13,983,81

5

, not 1 to 13,983,81

6

.

Odds for & against

Find expected value of an event

Maximizing & Minimizing

Ex: There are 4 aces in a standard deck of 52 cards.

Thus, if a single card is drawn from the deck, there are 4 ways of getting an ace (

favorable

) and 48 ways of not getting an ace (

unfavorable

).

The odds

in favor

of drawing an ace are

4

to

48

, or 1 to 12.

Note: The odds

against

drawing an ace are 48 to 4, or 12 to 1.

A horse named Lightning has a record of 73 wins and 4 losses. Based on this record, what is the probability of a win for this horse?

Here, f = 73 and u = 4... so P (w) = 73 / 73+4

Note: Since f = 73 and u = 4, the

odds

for Lightning to

win

73 races and lose 4 are 73 to 4. What are the odds against Lightning to win?

We often associate chance with betting. Suppose that a given event E has probability P(E) = f (favorable events)/n (number of outcomes) of occurring and P(not E) = u (unfavorable events)/n (number of outcomes) of not occurring.

If we now agree to pay $f if E does not occur in exchange for receiving u dollars if E does occur, then we can calculate our “expected average winnings” by multiplying P(E) (the approximate proportion of the times we win) by u (the amount we win each time).

Similarly, our losses will be P(not E) * f, because we lose $f dollars approximately P(not E) of the times.

If the bet is to be fair, the average net winnings should be 0.

Sometimes we wish to compute the expected value, or mathematical expectation, of a game.

Ex. If a woman wins $6 when she obtains a 1 in a single roll of a die and loses $12 for any other number, She will win $6 one-sixth of the time and she will lose $12 five-sixths of the time.

We then expect her to gain ($6)(1/6) - ($12)(5/6) = -$9 that is, to lose $9 per try on the average.

Statewide Insurance has determined the probability of paying various claim amounts as shown in the table. Find the expected value of a claim with Statewide Insurance.

Let a1 = 0, a2 = $500, a3 = $1000, a4 = $2000, a5 = $5000, a6 = $8000, a7 = $10,000 and p1, p2 and so on be the corresponding probabilities.

Note that to be fair the premium (amount you pay) for a policy based on this information should be $310

A die is rolled. A person receives double the number of dollars corresponding to the dots on the face that turns up. How much should a player pay for playing in order to make this a fair game?

Suppose you have two choices for a personal decision. Let us call these choices A and B. With choice A, you can make $20 with probability 0.24, $35 with probability 0.47, and $50 with probability 0.29.

With choice B you can lose $9 with probability 0.25, make nothing ($0) with probability 0.35, and make $95 with probability 0.40.

Make a tree diagram and determine what your decision should be if you want to maximize your profit.

What is the Expected value for choice A? For choice B?

12.1 Sampling, Frequency Distributions, & Graphs

12.2 Measures of Central Tendency: The Mean, Median, & Mode

12.3 Measures of Dispersion: The Range & Standard Deviation

12.4 The Normal Distribution: A Problem-Solving Tool

The science of collecting, organizing, and summarizing data (descriptive statistics) so that valid conclusions can be drawn from them (inferential statistics).

Sampling

Voting: Who will be the next governor in your state? What about the next president of the U.S.? How can we figure this out?

We can't ask every registered voter (target population) for whom s/he plans to vote, so analysts concentrate on a select, smaller number of people (sample population) to represent the entire group and then project the result to all registered voters.

In order for the conclusions reached to be valid, a simple random sample must be used.

The owner of the Latin Grill wants to determine if the items on his new menu satisfy the demands and tastes of his customers. He has a list of 1000 past customers who have visited the restaurant but because of budget and time constraints he cannot contact all of them, so he decides to make a simple random sample of 50 customers.

(a) What is the population? The 1000 previous customers

(b) What procedure can you use to select the simple random sample? Put their names in a hat

(c) What is the sample? The 50 selected

Suppose that the person conducting the survey selects the first 10 people who walk in and asks each of them if he or she owns a cell phone. Each person falls into one of two categories: yes (Y) or no (N). The responses are

1 Y, 2 N, 3 Y, 4 Y, 5 N, 6 Y, 7 N, 8 Y, 9 Y, 10 Y

1. What do you think the target population and the sample population are? Are they the same?

2. Do we have a simple random sample?

3. How can we summarize the data?

One is using a table or we can even make a picture (graph) of the results, as shown below. The table is called a frequency distribution; the graph is a histogram, which can be converted easily into a frequency polygon.

Frequency Distributions

The point of Stats: Statistical studies start by collecting data. In order to organize and summarize these data to detect any trends that may be present, we can use three types of tools: frequency distributions, histograms, and frequency polygons.

Let us look at a statistics problem that should interest a teacher and students, both of whom might wonder how well the students are learning a certain subject. Out of 10 possible points, the class of 25 students made the following scores:

This listing shows at once that there were some good scores and some poor ones, but because the scores are not arranged in any particular order, it is difficult to conclude anything else from the list.

A

frequency distribution

is often a suitable way of organizing a list of numbers to show what patterns are present. Create a chart listing all scores and making tallies of each one in the data. Add them up.

It is now easier to see that a score of 8 occurred more times than any other number. This score was made by what percent of the students?

Make a frequency distribution for the following data and represent each score as a percentage. Ages of students in a freshman class: 18, 20, 18, 17, 19, 23, 45, 21, 22, 19, 18, 20, 37, 18, 18, 20, 21, 19, 23, 18, 20, 19, 17.

Histograms & Frequency Polygons

It is also possible to present the information contained in our first frequency distribution table by means of a special type of graph called a histogram, consisting of vertical bars with no space between bars.

From the histogram we can construct a

frequency polygon

(or line graph) by connecting the midpoints of the tops of the bars.

Make sure to "tie the graph down" at the endpoints.

In a study of voter turnout in 20 cities with populations of over 100,000 in the United States, the following data were found:

Inspection of the data shows that the smallest number is 58.5 and the largest is 87.2. This time we go from 55 to 90, with the convenient class width of 5 units.

(a) Make a frequency distribution of the data on voting rate (r) using a class interval of 5% so that the classes will be 55 < r 60, 60 < r 65, . . . , 85 < r 90.

(b) Make a histogram and a frequency polygon of this distribution.

(c) In what percent of the cities was the voting rate greater than 80%?

(d) In what percent was the voting rate less than or equal to 70%?

Applications

An investment banker is studying the histograms below

(a) Which tract has the most owner-occupied homes?

(b) Which tract has the most vacant homes?

(c) If you are the banker, in which tract would you invest your money?

Unfortunately, the histograms only give the relative frequency (percent) of owner-occupied homes, so we have to use the table below.

Tongue Twister Averages

- Is there a relationship between the number of words in a tongue twister and the difficulty in reciting it? Below are several tongue twisters and the percentage of successful attempts out of 30 total attempts at reciting each.

What is the average number of words in each? It depends on what we mean by average.

The most commonly used measure of central tendency of a set of n numbers is the

mean

(the arithmetic average), which is obtained by adding all the numbers in the set and dividing by n. The mean of the number of words in the given tongue twisters is thus... ?

Another measure of central tendency is the

median

, the middle value of an ordered set of numbers (there are as many values above as below the median). Let us arrange the number of words in each twister in ascending order.

4, 4, 5, 5, 6, 6, 7, 8, 9, 10, 11, 13, 35

The final of the three types of central endency is the

mode

, the value occurring most often. We see that 4, 5, and 6 are modes for these numbers; they occur twice each.

So what is the average number of words in these tongue twisters? Either 4, 5, 6, 7, or 9.5.

Thus, it is possible for a set of numbers to have more than one mode or no mode at all. The mean and the mode are useful because they give an indication of a sort of center of the set.

For this reason, they are called measures of central tendency.

Have you been exercising lately? The following are 10 different activities with the corresponding hourly energy expenditures (in calories) for a 150-lb person:

(a) Find the mean of these numbers.

(b) Find the median number of calories spent in these activities.

(c) Find the mode of these numbers.

Practice

Alberto and Barney have just gotten back their test papers. There are 9 questions, and each one counted 10 points. There scores are as follows:

Person Question

1 2 3 4 5 6 7 8 9 Total

Alberto 10 7 10 7 7 10 9 10 2 72

Barney 10 8 10 7 7 7 10 7 7 73

Find all three central tendencies (Mean, Median, Mode) for each (Alberto and Barney) and then for both combined.

An investor bought 150 shares of Fly-Hi Airlines stock. He paid $60 per share for 50 shares, $50 per share for 60 shares, and $75 per share for 40 shares. What was his mean cost per share?

Range & SD

Ratings Deviations for Movies and TV

Which programs get the best television ratings, weekly series or movies?

The ratings for the 10 best series and 10best movies are shown to the right.

Range: series’ ratings from 18 to 24 (6 pts), movies’ ratings from 19 to 25 (6 pts).

Since the range is the same in both, you might look at the mean rating in each category.

But the mean is 21 in both cases. What else can you look at to try to determine the best air time for an advertiser?

There is a measurement that indicates how the data differ from the mean, and it is called the

standard deviation

(SD).

The standard deviation, like the range, is a measure of the spread of data.

To obtain the standard deviation of a set of numbers, start by computing the difference between each measurement and the mean, that is, as shown in Table 12.17 for the movie ratings data.

Unfortunately, if you add the values, you get a sum of 0. Therefore, you square each value before you do the addition and then arrive at a sum of 32.

If you were looking at the entire population of movies on TV, you would divide this number by the population size to get a type of “average squared difference” of ratings from the mean. However, this sample does not include the entire population, so, as a rule, divide instead by 1 less than the number in the sample (here, 10 – 1 = 9) to make the final value of the standard deviation a bit larger.

You then have in units of squared ratings points. What kind of unit is that?

To return to ratings points, you need to take the square root of

or

A number that describes how the numbers of a set are spread out, or dispersed, is called a

measure of dispersion

. A very simple example of such a measure is the range.

The most commonly used measure of dispersion, the standard deviation

Find the standard deviation of the TV series.

A consumer group checks the price of 1 dozen large eggs at 11 chain stores, with the following results:

Find the mean, median, mode, and standard deviation. What percent of the data are within 1 standard deviation from the mean?

The result is s = 4.3. To find the percent of the data within 1 standard deviation from the mean, we first find and

By examining the data, we see that 8 of the prices are between these two numbers.

8/11 = 0.727272... Thus, 73% of the prices are within 1 standard deviation from the mean price.