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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 COPIES OF ALL THE REVIEWS ARE AVAILABLE ON MY WEBPAGE.