Discrete Math
NCFinal Exam: Refresh your memory on …
 Matrix multiplication
o Dimension m x n * n x p is ok and will result in a matrix of dimension m x p; m x p * n x p is not ok
o Must be SQUARE to be raised to a power
 Graph Theory
o In any graph, the sum of the degrees of the vertices = (number of edges * 2)
o In any graph, the ODD vertices always happen in pairs (never an odd number of odd vertices)
o In a COMPLETE graph with N vertices (KN)…

The degree of each vertex is ( N  1)

The sum of the degrees is ( N )( N  1)

The number of edges is

The number of Hamilton Circuits or Paths is ( N  1) !

Complete graphs can be used to model problems like number of matches in a round-robin
( N )( N  1)
2
tournament, number of handshakes, number of pairwise comparisons, etc.
o Identify which perspective you are working with:

Street Sweeper = Every Edge = Euler Circuits/Paths

Connected, with 0 odd vertices = Euler Circuit/Closed Unicursal Tracing
o Can start anywhere and end up back at that point

Connected, with 2 odd vertices = Euler Path/Open Unicursal Tracing
o The 2 odd vertices are the start & stop points

Pizza Guy = Traveling Salesman = Every Vertex = Hamilton Circuits

Nearest Neighbor = Leap Frog from required vertex

Repetitive Nearest Neighbor = Leap Frog from EACH vertex; Choose best one; Write
circuit from required vertex

Cheapest Link = Jigsaw Puzzle: Sort pieces; Use vertex-only workspace; Add edges
but do NOT make degree of 3 OR a circuit; When (Number of Edges Chosen) =
(Number of Vertices – 1), close the Hamilton Circuit; Write circuit from required
vertex

Cable Company = Minimum Spanning Tree

Trees are “barely” connected
o Number of Edges = (Number of Vertices – 1); Redundancy = 0

Kruskal’s Algorithm = Jigsaw Puzzle: Sort pieces; Use vertex-only workspace; Add
edges but do NOT make a circuit (CAN have degree of 3); When (Number of Edges
Chosen) = (Number of Vertices – 1), you have the OPTIMAL minimum spanning
tree
 Measures of Location and Spread
o Location: Mean (average), Median (50th percentile), Percentiles
o Spread: Standard Deviation, IQR
o Five-number Summary, Quartiles
 Normal Distribution
o Line of Symmetry at the center = Mean (μ) = Median (M)

Points of Inflection are 1 Standard Deviation (σ) to the left and right of the Line of Symmetry


PL = μ – σ and PR = μ + σ
Quartiles are 0.675 Standard Deviations to the left and right of the Line of Symmetry

Q1 = μ - 0.675* σ and Q3 = μ + 0.675* σ
o z-values are used to Normalize the data (make the units standard)
( DataValue   )

z-value =

Negative z-values are to the left of the Line of Symmetry; Positive to the right

o 68 – 95 – 99.7 Rule: In a Normal Distribution…

68% of the data falls within 1 Standard Deviation of the Mean

95% of the data falls within 2 Standard Deviations of the Mean

99.7% of the data falls within 3 Standard Deviations of the Mean

There are virtually only 6 Standard Deviations separating the Minimum from the Maximum
 Probabilities
o Multiplication Rule: ____ x _____ x _____ etc. based on the number of choices for each position

CAN be used when Repeats are allowed

CANNOT be used if Order doesn’t matter
o Permutations & Combinations

CANNOT be used when Repeats are allowed

r = number of positions; n = number of choices

Permutations for when order DOES matter (count them all)
o

n Pr 
n!
(n  r )!
Combinations for when order DOES NOT matter (take out the duplicates; divide by
the number of positions factorial)
o
nCr 
n!
(n  r )! r!
(dividing by r! takes out the duplicates)
o EXPERIMENTAL Probability of Event E =
(Number of Occurrences of the Event) ÷ (Number of Trials in the Experiment)
o THEORETICAL Probability of Event E =
(Number of Outcomes that “fit” the criteria of the Event) ÷ (Total Number of Outcomes)
o Probabilities are between 0 and 1

Pr (Impossible Event) = 0

Pr (Certain Event) = 1
o Pr (Complementary Event) = 1 – Pr (Event)
o With INDEPENDENT Events (no overlapping):

Pr (A and B) = Pr (A) * Pr (B)

Pr (A or B) = Pr (A) + Pr (B)
o Odds:

Pr (Event) = good ÷ bad

Odds of Event = good “to” bad

Odds against Event = bad “to” good
o With Events that are NOT Mutually Exclusive (they overlap), use Venn Diagrams
o Tree Diagrams

Branches from a single node all add up to 1

Probability of a pathway = branch * branch * branch etc along the pathway

Sum of all complete pathways = 1
o Expected Value = Sum of all Outcomes * Each Outcome’s Probability

Represents the AVERAGE outcome (“ in the long run”)
o Binomial Expansion: (x + y) n

Number of terms = n + 1

Coefficients from Pascal’s Triangle

x powers go from n down to 0; y powers go up from 0 to n

x power + y power = n

(x + y) n has all positive terms; (x – y) n has alternating terms: positive, negative, positive,
etc.

To find the mth term of (x + y) n

(n – m) = z

n Cz
* xz * y(n – z)
 Apportionment
o “Seats” are divided by “States” based on “Population” (“what”, “who”, “how”)
o Standard Quota = Population ÷ Standard Divisor
o Standard Divisor = Total Population ÷ Number of Seats
o Hamilton’s Method = Method of Largest Remainder
o JAW DUC
 Election Theory
o Majority = More than half; Plurality = Most
o Plurality Method: Most 1st place votes wins
o Borda Count Method: Points are earned based on preference with 1 for last, 2 for second to last,
etc.; Points for each candidate are totaled; Highest wins
o Plurality-with-Elimination Method (we called this “Eliminate Until Majority”): Eliminate the
candidate with the fewest 1st choice votes; Recount; Continue until a majority candidate wins

Also known as Hare’s Method or Instant Runoff Voting
o Method of Pairwise Comparisons: Compare each pair of candidates in head-to-head comparisons;
Award 1 point for a win or ½ point to each for a tie; Total up points; Highest wins

Also known as Copeland’s Method

An “undefeated” candidate is called the Condorcet Candidate
o Ranking Methods: Used to find the rankings, not just the winner

Extended: Do the method ONCE and then rank them (very intuitive)

Recursive: RE-Do the method over and over, removing the winner each time (very tedious –
especially Recursive Borda Count as new point values are needed)
 Voting Power
o [q : w1, w2, w3, … wn]

q (the “quota”) = the Number of Votes needed to Pass a Motion

wn (the “weight of Player n”) = the Number of Votes Player n Controls

Quota should always be more than ½ the total votes and less than all of them

Banzhaf Power Index: Based on how often a Player is CRITICAL in a Winning Coalition

Shapley-Shubik Power Index: Based on how often a Player is PIVOTAL in a Sequential
Coalition
 Fair Division
o Discrete Loot:

Method of Markers

Method of Sealed Bids
o Continuous Loot:

Divider-Chooser Method

Lone Divider Method

Last Diminisher Method
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