Summary 3 Discrete Probability (sections 5.1 - 5.2) Experiments, outcomes, and sample space Use counting techniques to determine sample space p(s) for each sS 0  p(s)  1 for each sS sS p(s) = 1 If all outcome are equally probable, then p(s) = 1/|S| Events and event probability E  S, P(E) = |E|/|S| or P(E) = sE p(s) Use counting techniques to determine samples in E Complementary event: P(E) = 1 – P(-E).

Conditional probability P(E|F) Definition: probability of E, given F (or in subspace F S) Relation to joint probability P(E|F) = P(EF)/P(F) or P(E|F) = |EF|/|F| P(EF) = P(E|F)P(F) = P(F|E)P(E) Inclusion-exclusion rule: P(EF) = P(E) + P(F) – P(EF) Independence Events E and F are independent of each other if P(E|F) = P(E) (E’s probability not depending on F) P(EF) = P(E) + P(F) –P(E)P(F)

Bernoulli Trials Experiment with two outcomes, s and f, p = P(s), q = P(f) = 1– p (therefore p + q = 1) n independent trials with k s (and n – k f) C(n, k)pkqn-k

Relations (sections 7.1 – 7.5) Definitions. (Binary) relation R from A to B: R  AB: ordered pairs Relation R on A: R  AA n-ary relation R  A1A2…An Functions are special cases of relations. Properties of relations Reflexive/irreflexive Symmetric/asymmetric/antisymmetric Transitive Properties reflected in graph and matrix representations Combining relations: R1  R2, R1  R2, R1 – R2, SR.

Equivalence relations Definition: reflexive, symmetric and transitive. Equivalence class: [a]R For all b, if bRa, then b  [a]R aRb iff [a] = [b] iff [a]  [b]   not aRb iff [a]  [b] iff [a]  [b] =  Partition of a set: (i) Ai   for iI (ii) Ai  Aj = , if i  j (iii) iI Ai = S All equivalence classes of R on A partition A.

Representing relations Set of ordered pairs: {(a, b)| aRb for all (a, b)  AB} 0 – 1 matrix mij = 1, if (ai, bj)R, and mij = 0, if (ai, bj)R. Directed graph: (V, E) V = A  B ( or V = A if R is on A) (a, b) E iff (a, b) R Be able to convert between the three representations

Graphs (sections 8.1 – 8.4) Definitions. Simple, undirected and directed graphs. Degree (indegree, outdegree) of vertex Loop, isolated and pendant vertices Special graphs Complete, cycle, bipartite, and complete bipartite, n-cube Subgraph Adjacency For undirected graphs:vV deg(v) = 2|E| For directed graphs: vV deg-(v) = vV deg+(v) = |E|

Representing graphs Adjacency matrix aij = 1 if {vi, vj} is an edge of G; and aij = 0 otherwise. Connectivity Path, path length, simple path, circuit A (undirected) graph is connected if there is a path between any pair of vertices Strong and weak connected directed graphs Connected components of a graph

Types of Questions Conceptual Problem solving Proofs Definitions of terms True/false Simple questions Problem solving Work with small concrete example problems Proofs Simple theorems or propositions Possible proof methods Induction Directed proof Proof by contradiction