Chris Morgan, MATH G160 January 9, 2012 Lecture 1 Chapter 4.1, 4.2, & 4.3: Set Theory, Introduction to Probability

Sample Space Random Experiment: an action which produces well-defined but unpredictable outcomes Sample Space Ω: the set of all possible outcomes for a random experiment, with Ω representing the entire sample space, and ω representing each element (outcome) in that sample space Sample space can be finite or infinite Example: When flipping a coin once, the set of all possible outcomes: Ω = {Head, Tail} When flipping a coin twice, the set of all possible outcomes: Ω = {{Head, Head}, {Head, Tail},{Tail, Head}, {Tail, Tail}}

Sets and Elements Set (S) is a collection of distinct objects, also called a sample space Element is a single value from a set and can be qualitative or quantitative Examples? S = {Head, Tail); two elements S = {1, 2, 3, 4, 5, 6}; six elements Set of sets: S = {{Head, Tail}, {Head, Head}, {Tail, Tail}}

A few important Sets = collection of real numbers = collection of rational (can be viewed as a fraction) numbers = collection of integers = collection of positive integers Note the difference between () and []. () are called open intervals whereas [ ] are called closed intervals. If your interval contains infinity then it is considered unbounded (no infinity implies bounded). Also, conventionally, if you use infinity or negative infinity as one of your end points, then you use a ( with it. Ex: (-inf, 5] or (-inf, 5) depending on whether you want to include 5.

Subsets Subset is a portion of a set, also called an event - It is a sub-collection of the elements of some larger set Examples? {Head} is a subet of S = {Head, Tail) A = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} A is a set with all letters of the English alphabet The letter “a” is an element of A; denoted: The set V= {a, e, i, o, u} is a subset of A; denoted: S = {1, 2, 3, 4, 5, 6} {1, 2, 3} is a subset of S {2,7} is not a subset of S

Subsets (cont) S = {Purdue, Michigan, Wisconsin, Minnesota, IU} S is a set containing five schools in the Big Ten Conference The school “Purdue” is an element in the set S, denoted by: The school “Illinois” is not an element in the set S, denoted by: The set B = {Minnesota, Wisconsin} is a subset of S, denoted by: The set which contains nothing is called an empty set,

Event Event (A): a subset of the sample space Example: Roll a dice once, get a result less than 4, A = {1, 2, 3} is an event Example: Flip a fair coin twice, get at least one tail, A = {{Head, Tail}, {Tail, Tail}} Complement of an event (A c ): a set that includes ALL sample points that are not in A Example: Roll a dice once, A = {1, 2, 3}, then Ac = {4, 5, 6} Example: Flip a fair coin, A = {{Head, Tail}, {Tail, Tail}}, A c = {{ Head, Head}}

Set Operations Given two sets A and B we can form their union, denoted by: A B = {x: x A or x B} This is the collection of elements in A or B Note: Whenever we use the word “or” it is to be taken in the logical sense and not in the exclusive sense; meaning if you were given two options for an action when we say Option I or Option II we mean one or the other or both

Venn Diagrams… … your new best friend.

B AB C A

Set Operations Given two sets A and B we can form their intersection, denoted by: A B = {x: x A and x B} This is the collection of elements in A and B

B AB C A

Set Operations Given a set A we can find its complement, denoted by: A c = {x: x A} The compliment of A is the collection of elements which are not in A AB

Set Operations Given a set A we can find its power set, denoted by: This is the collection of all possible subsets of A. It is a set whose elements are again sets, i.e. a set of sets. For a set containing n elements the corresponding power set will contain 2 n elements.

Examples of Set Operations Find the following:

Probability Axioms A probability is a real number between 0 and 1 which represent the likelihood an event will occur when a random experiment is conducted - closer to 1 the more likely the event will happen - closer to 0 the more unlikely an event is to occur A probability measure P, i.e. P(Heads) is a function which takes an event as an argument and returns a probability corresponding the event 1.P (Ω) = 1 & P( ) = 0 2. Given any two events A and B such that: 3. Given any two events A and B such that:

Probability Axioms 4. If we are given a random experiment with finitely many equally likely outcomes then the probability of any event is just the number of outcomes corresponding to the event divided by the number of outcomes in the sample space. In other words, for any event or equivalently we have: where N(A) equals the number of outcomes in the event A

Definitions Mutually Exclusive Events: two events that cannot occur at the same time; disjoint or NO intersection Collectively Exhaustive: at least one of the events must occur when the random experiment is performed (rolling a die) A and B are collectively exhaustive if: A U B = S Is “S” mutually exclusive or collectively exhaustive?