Spaces Outline
Sample and Probability Lecture 9, STAT 2246
Julien Dompierre
Dpartement de mathmatiques et dinformatique e e Universit Laurentienne e
25 janvier 2007, Sudbury
Julien Dompierre
1
Outline
Set Theory Probability Theory
Outline
1
Sample Spaces and Probability Set Theory Probability Theory
Julien Dompierre
2
Outline
Set Theory Probability Theory
Probability
A cynical person once said, The only two sure things are death and taxes. This philosophy no doubt arose because so much in peoples lives is aected by chance. From the time a person awakes until he or she goes to bed, that person makes decisions regarding the possible events that are governed at least in part by chance. For example, should I carry an umbrella to work today? Will my car battery last until spring? Should I accept that new job?
Julien Dompierre
3
Outline
Set Theory Probability Theory
Probability (p. 179)
Probability as a general concept can be dened as the chance of an event occurring. Many people are familiar with probability from observing or playing games of chance, such as card games, slot machines, or lotteries. In addition to being used in games of chance, probability theory is used in the elds of insurance, investments, and weather forecasting and in various other areas. Finally, probability is the basis of inferential statistics. For example, predictions are based on probability, and hypotheses are tested by using probability.
Julien Dompierre
4
Outline
Set Theory Probability Theory
Outline
1
Sample Spaces and Probability Set Theory Probability Theory
Julien Dompierre
5
Outline
Set Theory Probability Theory
Set and Elements
A set is a collection of objects. The number of objects in a set can be nite or innite. The only common property of these objects is to belong to the set. The objects in the set are also denoted elements of the set.
Julien Dompierre
6
Outline
Set Theory Probability Theory
Membership
One writes a A to denote that a is an element of (or belongs to) the set A. One writes a A to denote that a is not an element of (or does / not belong to) the set A.
Julien Dompierre
7
Outline
Set Theory Probability Theory
Set Representation
The most common method to write a set is to list all its elements inside braces. For example, the set V of vowels is denoted V = {a, e, i, o, u, y }.
Julien Dompierre
8
Outline
Set Theory Probability Theory
Set Equality
Two sets are equal if and only if they contains the same elements. Example: The sets {1, 3, 5} and {3, 5, 1} are equal.
Julien Dompierre
9
Outline
Set Theory Probability Theory
Venn Diagram
Venn diagram is a graphical representation of a set. The rectangle represents the universal set of all objects analysed. Inside the rectangle, circles are used to represent sets. We use points to denote specic elements of a set. This is the Venn diagram of the set of all vowels: U V a i y e o u
Julien Dompierre
10
Outline
Set Theory Probability Theory
Empty Set
There exists a special set that contains no element. This set is called the empty set. The empty set is denoted by or by { }.
Julien Dompierre
11
Outline
Set Theory Probability Theory
Subset
The set A is a subset of the set B, denoted A B, if and only if all the elements of A also belong to the set B. The empty set is a subset of all sets, i.e., S, for all sets S. Every set is a subset of itself, i.e. S S, for all sets S. This Venn diagram shows a set A subset of the set B. U B A
Julien Dompierre
12
Outline
Set Theory Probability Theory
Cardinality
Let S be a set. If S has exactly n elements, where n is a non negative integer, then S is called a nite set and n denotes the cardinality of S. The cardinality of the set S is denoted by n(S) (or by |S| in many textbooks). The cardinality of the empty set is 0, i.e., n() = 0.
Julien Dompierre
13
Outline
Set Theory Probability Theory
Set Union
Let A and B be two sets. The union of sets A and B, denoted A B, is the set that contains the elements that belong to the set A, or the set B, or both. A B = {x | x A or x B}.
U A B
Julien Dompierre
14
Outline
Set Theory Probability Theory
Set Intersection
Let A and B be two sets. The intersection of sets A and B, denoted A B, is the set that contains the elements that belong to both sets A and B. A B = {x | x A and x B}.
U A B
Julien Dompierre
15
Outline
Set Theory Probability Theory
Mutually Exclusive Sets
Two sets are disjoint or mutually exclusive if their intersection is empty. U A B
Julien Dompierre
16
Outline
Set Theory Probability Theory
Set Dierence
Given sets A and B, the dierence A B (sometimes denoted as A\B) is dened to be the set of elements of A that are not in B. In other words, A B = {x | x A and x B}. /
U A B
Julien Dompierre
17
Outline
Set Probability Theory Theory
Set Complement
The complement of a set A is the set of all elements of the universal set U that are not in A. The complement of A is denoted by A (other pieces of notation used by authors are Ac and A ). A = {x | x A}. / U A
Note: The union of a set with its complement gives the universal set, i.e. A A = U.
Julien Dompierre 18
Outline
Set Theory Probability Theory
Outline
1
Sample Spaces and Probability Set Theory Probability Theory
Julien Dompierre
19
Outline
Set Theory Probability Theory
Experiment and Outcome (p. 179)
A probability experiment (or experiment) is any action or process that leads to well-dened results called outcomes. An outcome is the result of a single trial of a probability experiment.
Julien Dompierre
20
Outline
Set Theory Probability Theory
Sample Space (p. 179)
The sample space of an experiment, denoted by S, is the set of all possible outcomes of that experiment. Experiment Sample space Toss a coin S = {Head, Tail} Toss two coins S = {(H, H), (H, T ), (T , H), (T , T )} Roll a die S = {1, 2, 3, 4, 5, 6} Answer a true/false question S = {True, False} Have a baby S = {Boy, Girl} Have two babies S = {(B, B), (B, G ), (G , B), (G , G )}
Julien Dompierre
21
Outline
Set Theory Probability Theory
Sample Space (p. 180)
The experiment is rolling two dice. The sample space S of all possible outcomes is
Julien Dompierre
22
Outline
Set Theory Probability Theory
Sample Space (p. 180)
The experiment is drawing one card from an ordinary deck of card. Since there are 4 suits (hearts, clubs, diamonds, and spades) and 13 cards for each suit (ace through king), there are 52 outcomes in the sample space S:
Julien Dompierre
23
Outline
Set Theory Probability Theory
Tree Diagram (p. 181)
A tree diagram is a device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine the sample space S of all possible outcomes of a probability experiment.
Julien Dompierre
24
Outline
Set Theory Probability Theory
Event (p. 181)
An event is any subset E of outcomes contained in the sample space S. An event is said to be a simple event if it consists of exactly one outcome. An event is said to be a compound event if it consists of more than one outcome.
Julien Dompierre
25
Outline
Set Theory Probability Theory
Classical Probability (p. 182)
Classical probability uses sample spaces to determine the numerical probability that an event will happen. One does not actually have to perform the experiment to determine that probability. We suppose the events are equally likely, i.e., they have the same probability of occurring. Denition The probability P(E ) of an event E S is dened as P(E ) = n(E ) n(S)
Julien Dompierre
26
Outline
Set Theory Probability Theory
Probability Rules (p. 184)
Th...