Professor J' P' Thomas

1
PROBLEM SOLVING

• List everything that you know about the problem.

• Identify the question or unknown.

• List any underlying formulae.

• Draw a picture.

• Solve the problem.
• (Dont take short cuts or skip any steps).

2
VOTING METHODS
Plurality Method the most votes wins.
Plurality Method with a Runoff winner needs
more than 50 of votes
Preference Ranking Ordering candidates by
preference.
Bordas Method a scoring system.
Head to Head Comparisons the most votes
between 2 candidates.
Approval Method highest number of approval
votes.
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PREFERENCE RANKING
Number of Voters
Plurality
4
2
3
2
By Plurality Method Tuesday Wins
4
PREFERENCE RANKING
Number of Voters
4
By Plurality with a Runoff Thursday Wins
5
BORDAS METHOD
Count
10 11 9
1 2 3
2 1 3
2 3 1
3 2 1
3 2 1
2 3 1
2 1 3
1 2 3
3 2 1
2 1 3
1 2 3
4 6 2
Winner by Borda Count - Horowitz
Winner by Preference Vote Horowitz Taylor tied.
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HEAD TO HEAD COMPARISON
2 3
3 2
3 2
Since Horowitz beat both Coleman and Taylor in a
Head to Head Comparison, Horowitz is a Condorcet
Winner!
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APPROVAL VOTING
6 8 4 1 6
Approval voting winner Alvarez
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COMPARISON OF VOTING METHODS
Preference Voting June July Tie.
By Borda Count June Wins.
Approval Voting June Wins.
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IDEAL VOTING SYSTEM
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EXPONENTS

LAWS OF EXPONENTS

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EXPONENTIAL FUNCTION
y b x
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SOLVING EXPONENTIAL FUNCTIONS ALGEBRAICALLY
SOLVE BY
IF
Equal Bases
Bases are equal
Inverse Function y b x ? x log b y
Base is 10 or e
Log of Function
In any case
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PROPERTIES OF LOGARITHMS
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INTEREST FORMULAE
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PERIODIC PAYMENTS
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AMORTIZATION SCHEDULE
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PATHS AND CIRCUITS
Eularian Paths and Circuits
Paths through a network where every edge is used
only once.
Hamiltonian Paths and Circuits
Paths through a network where every vertex is
used only once.
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PATHS
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EURLARIAN PATHS AND CIRCUITS
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EULARIZATION
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NEWSPAPER ROUTE EULARIZATION
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HAMILTONIAN PATHS AND CIRCUITS
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TRAVELING SALESMAN PROBLEM
Greedy Algorithm
Nearest Neighbor Algorithm
4 8 11 2 10 35
2 4 5 8 11 30
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TRAVELING SALESMAN PROBLEM
Nearest Neighbor Algorithm
Greedy Algorithm
2 3 7 4 5 5 26
2 3 10 9 4 4 32
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SPANNING TREES
Spanning Tree
Subgraph that is not a tree
Subgraph that is a tree, But not a Spanning Tree
Subgraph that is not a tree Not connected.
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PRIMS ALGORITHM
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REGULAR POLYGONS
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CLASSIFICATION OF TRIANGLES
BY ANGLES
BY SIDES
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TYPES OF QUADRILATERALS
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MODULAR ARITHMATIC
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BASE 10 NOTATION
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DIVISIBILITY TESTS
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ROUND ROBIN SCHEDULING
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CRYPTOLOGY
35
INVERSE MOD 26
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HILL CIPHER
ENCRYPTION
DECRYPTION
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INVERSE MOD 26
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PROBABILITY DEFINITIONS
Probability mathematical estimate of the
likelihood that some event will occur.
Empirical Probability
frequency of occurrence of an event as
determined by an experiment.
Theoretical Probability
frequency of occurrence of an event as
determined mathematically
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PROBABILITY DEFINITIONS
Experiment an examination to estimate the
likelihood that some event will occur.
Event subset of outcomes of an experiment.
Outcome result of an experiment.
Sample Space set of all possible outcomes.
Equal Likelihood each event in the sample space
has the same probability of occurring.
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PROBABILTY EQUATIONS
Empirical Probability P(E) number of times
event E occurred . number of
times experiment performed
Theoretical Probability P(E) number of
favorable outcomes . total number of
possible outcomes
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BASIC PROPERTIES OF PROBILITIES
1. The probability of an event is always between
0 and 1 inclusive. 0 lt P(E) lt 1
2. The probability of an event that cannot occur
(impossible event) is 0.
3. The probability of an event that must occur
(certain event) is 1.
4. The sum of the probabilities of all possible
outcomes of an experiment is 1. SP(E) 1
5. The probability of an event not occurring is
1 minus the probability of its occurring.
P(E) 1 P(E) or P(E) 1 P(E)
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LAWS OF PROBABILITIES
P(A or B) P(A ? B) P(A) P(B) P(A ? B)
P(A or B) A and B Mutually Exclusive P(A ? B)
P(A) P(B)
P(A given B) P(AB)P(A ? B) P(B)
P(A and B) P(A ? B) P(AB) P(B)
P(A and B) A and B Independent P(A ? B) P(A)
P(B)
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EXPECTED VALUE