53 terms

Finite Math 105 Final

Math Math Math and more Math
STUDY
PLAY
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
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}
Set-Builder Notation
Used to express common properties of the elements in a set
Ex. B={x/x has property P}
Element
Set cannot be an element of a set.
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( )
Empty Set (Null Set)
Set that contains no element
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
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
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
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"
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"
Disjoint Sets
Sets that have no elements in common
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
Procedure to Change a Fraction to a Percent Number
1. Divide numerator by denominator (obtain quotient)
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
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) )
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
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
Complement rule for probability
For any event 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
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)
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
Fair price
Amount to be paid that results in an expected value of $0
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
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
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) )
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