Conjecture or Hypothesis

Prediction made based on specific observations

Inductive Reasoning

The process of reasoning to a general conclusion through observation of specific cases

Deductive Reasoning

The process of reasoning that is used to prove a conjecture

Counterexample

Special case that satisfies the conditions of a conjecture but gives different results

Natural Numbers or Counting Numbers

Any number starting from the number 1

Problem Solving Guidelines

1. Understand the problem

2. Devise a plan to solve the problem

3. Carry out the plan

4. Check the results

2. Devise a plan to solve the problem

3. Carry out the plan

4. Check the results

Set

Collection of objects. The objects are called elements or members of the set

Well Defined Set

Set of which content can be clearly determined

Roster Form

List the elements of the set inside a pair of brackets

Ex. A={Mon, Tues, Wed, Thur, Fri, Sat, Sun}

Ex. A={Mon, Tues, Wed, Thur, Fri, Sat, Sun}

Set-Builder Notation

Used to express common properties of the elements in a set

Ex. B={x/x has property P}

Ex. B={x/x has property P}

Element

Set cannot be an element of a set.

Any number inside a set

Symbol is an ∈

Any number inside a set

Symbol is an ∈

Finite Set

A set that contains no element OR the number of elements in the set is a natural number (you can count the number of elements)

Equivalent Set

Set A is equivalent to set B if and only if A and B have the same cardinality (n(A)=n(B))

Universal Set (universe, U)

Set that contains all elements pertaining to a discussion

Cardinal Number (cardinality)

Number of elements in a set

Notation is n( )

Notation is n( )

Empty Set (Null Set)

Set that contains no element

Symbol is { } or ᴓ

C={ᴓ} is not an empty set n(C)=1

Symbol is { } or ᴓ

C={ᴓ} is not an empty set n(C)=1

Subset

Set A is a subset of set B if all elements of A are elements of B

Symbol is ⊆

# of subsets= 2 to the nth power

Symbol is ⊆

# of subsets= 2 to the nth power

Proper Subset

Set A is a proper subset of set B if A is a subset of B (A⊆B) and A is "smaller than" B

The notation is ⊂

# of proper subsets= 2 to the nth power subtracted by 1

The notation is ⊂

# of proper subsets= 2 to the nth power subtracted by 1

Complement of Set A

Set of all elements in the universal set but not in A

The notation is A ́ (read A-prime)

For any set A, (A ́) ́=A

The notation is A ́ (read A-prime)

For any set A, (A ́) ́=A

Intersection of Two Sets A and B

Set of all elements that belong to both set A AND set B

The notation is A ∩ B (read A "inter" B)

Also means "in common"

The notation is A ∩ B (read A "inter" B)

Also means "in common"

Union of Two Sets A and B

Set of all elements belonging to set A, OR set B, or to both sets

The symbol is ∪

Also means "combine"

The symbol is ∪

Also means "combine"

Disjoint Sets

Sets that have no elements in common

A∩B= ∅

For any set A, A∩A ́= ∅ or { } A and A ́ are disjoint

A∩B= ∅

For any set A, A∩A ́= ∅ or { } A and A ́ are disjoint

Union Rule for Sets

n(A∪B)=n(A)+n(B)-n(A∩B)

Venn Diagrams

Start with intermost

Procedure to Change a Decimal to Percent

1. Multiply by 100

2. Add a percent sign

2. Add a percent sign

Procedure to Change a Fraction to a Percent Number

1. Divide numerator by denominator (obtain quotient)

2. Multiply by 100

3.Add percent sign

2. Multiply by 100

3.Add percent sign

Procedure to Change a Percent to a Decimal Number

1. Divide the number by 100

2. Remove the percent

2. Remove the percent

Percent Change

Amount in Latest Period minus Amount in Previous Period divided by Amount in Previous Period

Percent Markup (on Dealers Cost)

Selling Price minus Dealer's Cost divided by Dealer's Cost

Experiment

Activity or occurrence with an observable results

Trial

A repetition of an experiment

Outcomes

Possible results of an experiment

Sample Space (S)

Set of all possible outcomes of an experiment

Event

Sub-collection of the outcomes of an experiment

Empirical Probability (of and event E)

Relative frequency of occurrence of that event

# of times event E occurred / # of times experiment is performed (frequency-count........relative frequency-percent (in decimal) )

# of times event E occurred / # of times experiment is performed (frequency-count........relative frequency-percent (in decimal) )

Law (of large numbers)

The relative frequency over the long run is more accurately predictable

Theoretical probability

Obtained through the study of possible outcomes that can occur for a given experiment

Equally likely outcomes

Outcomes that have given the same chance of occurring

P(E)=# of outcomes favorable to E \ # of possible outcomes

P(E)=# of outcomes favorable to E \ # of possible outcomes

Impossible event

If an event cannot occur its probability is 0

Certain event

If an event always occurs its probability is 1

Some important facts about probability

a) impossible event

b) certain event

c) For any event 0≤P(E)≤1 ( every probability is a number between 0 and 1 inclusive)

d) for any given experiment the sum of the probabilities of all possible outcomes is 1

b) certain event

c) For any event 0≤P(E)≤1 ( every probability is a number between 0 and 1 inclusive)

d) for any given experiment the sum of the probabilities of all possible outcomes is 1

Complement rule for probability

For any event A

P(A)+P(not A)=1 P(not A)=1-P(A)

P(A)+P(not A)=1 P(not A)=1-P(A)

Compute probabilities from odds

Then odds of event E are a to b

P(E)=a \ a+b P(not E)=b \ a+b

P(E)=a \ a+b P(not E)=b \ a+b

Compute odds from probabilities given P(E)

Odds in favor of E= P(E) \ P(not E)

Odds against E= P(not E) \ P(E)

Odds against E= P(not E) \ P(E)

Expected value

Used to determine the (expected) result of an experiment or business venture over the long run

Expected value formula

P1:probability that event 1 will occur

A1:amount won/lost if event 1 occurs

A1p1+A2p2...+Anpn

A1:amount won/lost if event 1 occurs

A1p1+A2p2...+Anpn

Fair price

Amount to be paid that results in an expected value of $0

Expected value+cost to play

Expected value+cost to play

Sample point

Each individual outcome in the sample space

Multiplication (or counting) principle

If a first experiment can be performed in M distinct ways, and a second experiment can be performed in N distinct ways, then the sequence of the two experiments in that order can be performed in M•N ways

In general (complement rule)

P(at least once)=1-P(none)

OR problem

In probability event A OR B occurs if event A occurs B occurs or both A and B occur

P(A OR B)=P(A)+P(B)-P(A and B)

Also called addition formula or union rule for probability

OR-Union-U-combine

P(A OR B)=P(A)+P(B)-P(A and B)

Also called addition formula or union rule for probability

OR-Union-U-combine

Mutually exclusive

Two events A and B are mutually exclusive if it is impossible for both events to occur simultaneously

P(A and B)=P(A intersection B)=0

P(A and B)=P(A intersection B)=0

AND problem

In probability event A and B occurs when event A and B occur

P(A AND B)=P(A)•P(B)

(Always assume that event A has happened when computing P(B) )

P(A AND B)=P(A)•P(B)

(Always assume that event A has happened when computing P(B) )