week 21 Basic Set Theory A set is a collection of elements. Use capital letters, A, B, C to denotes sets and small letters a 1, a 2, … to denote the elements of the set. Notation: means the element a 1 is an element of the set A A = {a 1, a 2, a 3 }. The null, or empty set, denoted by Ф, is the set consisting of no points. Thus, Ф is a sub set of every set. The set S consisting of all elements under consideration is called the universal set.

week 22 Relationship Between Sets Any two sets A and B are equal if A and B has exactly the same elements. Notation: A=B. Example: A = {2, 4, 6}, B = {n; n is even and 2 ≤ n ≤ 6} A is a subset of B or A is contained in B, if every point in A is also in B. Notation: Example: A = {2, 4, 6}, B = {n; 2 ≤ n ≤ 6} = {2, 3, 4, 5, 6}

week 23 Venn Diagram Sets and relationship between sets can be described by using Venn diagram. Example: We toss a fair die. What is the universal set S? …

week 24 Union and Intersection of sets The union of two sets A and B, denoted by, is the set of all points that are in at least one of the sets, i.e., in A or B or both. Example 1: We toss a fair die… The intersection of two sets A and B, denoted by or AB, is the set of all points that are members of both A and B. Example 2: The intersection of A and B as defined in example 1 is …

week 25 Properties of unions and intersections Unions and intersections are: Commutative, i.e., AB = BA and Associative, i.e., Distributive, i.e., These laws also apply to arbitrary collections of sets (not just pairs).

week 26 Disjoint Events Two sets A and B are disjoint or mutually exclusive if they have no points in common. Then. Example 3: Toss a die. Let A = {1, 2, 3} and B = {4, 5}.

week 27 Complement of a Set The complement of a set, denoted by A c or A’ makes sense only with respect to some universal set. A c is the set of all points of the universal set S that are not in A. Example: the complement of set A as defined in example 3 is… Note: the sets A and A c are disjoint.

week 28 De Morgan’s Laws For any two sets A and B: For any collection of sets A 1, A 2, A 3, … in any universal set S

week 29 Finite, Countable Infinite and Uncountable A set A is finite if it contains a finite number of elements. A set A is countable infinite if it can be put into a one-to-one correspondence with the set of positive integers N. Example: the set of all integers is countable infinite because … The whole interval (0,1) is not countable infinite, it is uncountable.

week 210 The Probability Model An experiment is a process by which an observation is made. For example: roll a die 6 times, toss 3 coins etc. The set of all possible outcomes of an experiment is called the sample space and is denoted by Ω. The individual elements of the sample space are denoted by ω and are often called the sample points. Examples... An event is a subset of the sample space. Each sample point is a simple event. To define a probability model we also need an assessment of the likelihood of each of these events.

week 211 σ – Algebra A σ-algebra, F, is a collection of subsets of Ω satisfying the following properties:  F contains Ф and Ω.  F is closed under taking complement, i.e.,  F is closed under taking countable union, i.e., Claim: these properties imply that F is closed under countable intersection. Proof: …

week 212 Probability Measure A probability measure P mapping F  [0,1] must satisfy For, P(A) ≥ 0. P(Ω) = 1. For, where A i are disjoint, This property is called countable additivity. These properties are also called axioms of probability.

week 213 Formal Definition of Probability Model A probability space consists of three elements (Ω, F, P) (1) a set Ω – the sample space. (2) a σ-algebra F - collection of subsets of Ω. (3) a probability measure P mapping F  [0,1].

week 214 Discrete Sample Space A discrete sample space is one that contains either a finite or a countable number of distinct sample points. For a discrete sample space it suffices to assign probabilities to each sample point. There are experiments for which the sample space is not countable and hence is not discrete. For example, the experiment consists of measuring the blood pressure of patients with heart disease.

week 215 Calculating Probabilities when Ω is Finite Suppose Ω has n distinct outcomes, Ω = {ω 1, ω 2,…, ω n }. The probability of an event A is In many situations, the outcomes of Ω are equally likely, then, Example, when rolling a die for i = 1, 2, …, 6. In these situations the probability that an event A occurs is Example:

week 216 Rules of Probability for all Corollary: The probability of the union of any two events A and B is Proof: … If then, Proof:

week 217 Inclusion / Exclusion formula: For any finite collection of events Proof: By induction