1. 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).

2. 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)

3. 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

4. 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.

5. 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.

6. 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

7. 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|

8. 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

9. 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