Basic Concepts and Approaches EMIS 7370 STAT 5340 Department of Engineering Management, Information and Systems Probability and Statistics for Scientists and Engineers Probability- Basic Concepts and Approaches Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering

Probability-Basic Concepts and Approaches Basic Terminology & Notation Basic Concepts Approaches to Probability Axiomatic Classical (A Priori) Frequency or Empirical (A Posteriori) Subjective

Basic Terminology Any well-defined action. It is any action or process Definition – Experiment Any well-defined action. It is any action or process that generates observations. Definition - Outcome The result of performing an experiment

Basic Terminology Definition - Sample Space The set of all possible outcomes of a statistical experiment is called the sample space and is represented by S. Remark: Each outcome in a sample space is called an element or a member of the sample space or simply a sample point.

Example An experiment consists of tossing a fair coin three times in sequence. How many outcomes are in the sample space? List all of the outcomes in the sample space.

Example An biased coin (likelihood of a head is 0.75) is tossed three times in sequence. How many outcomes are in the sample space? List all of the outcomes in the sample space.

Basic Terminology Definition - Event An event is the set of outcomes of the sample space each having a given characteristic or attribute Remark: An event, A, is a subset of a sample space, S, i.e., A  S.

Basic Terminology Definition - Types of Events If an event is a set containing only one element or outcome of the sample space, then it is called a simple event. A compound event is one that can be expressed as the union of simple events. Definition - Null Event The null event or empty space is a subset of the sample space that contains no elements. We denote the event by the symbol .

Operations With Events Certain operations with events will result in the formation of new events. These new events will be subsets of the same sample space as the given events. Definition - The intersection of two events A and B, denoted by the symbol A  B, or by AB is the event containing all elements that are common to A and B. Definition - Two events A and B are mutually exclusive if A  B = . Definition - The union of two events A and B, denoted by the symbol A  B, is the event containing all the elements that belong to A or to B or to both.

Operations With Events Definition - The complement of an event A with respect to S is the set of all elements of S that are not in A. We denote the complement of A by the symbol A´. Results that follow from the above definitions: A   = 0. A   = A. A  A´ =  A  A´ = S. S´ = . ´ = S. (A´) ´ = A. A A´ S Venn Diagram

Basic Concept For any event A in S, the probability of A occurring is a number between 0 and 1, inclusive, i.e., where and where Ø is the null event

Probability-Basic Questions (1) First, there is a question of what we mean when we say that a probability is 0.82, or 0.25. - What is probability? (2) Then, there is the question of how to obtain numerical values of probabilities, i.e., how do we determine that a certain probability is 0.82, or 0.25. - How is probability determined? (3) Finally, there is the question of how probabilities can be combined to obtain other probabilities. - What are the rules of probability?

Approaches to Probability Axiomatic Classical (A Priori) Frequency or Empirical (A Posteriori) Subjective

Axiomatic Approach Given a finite sample space S and an event A in S, we define P(A), the probability of A, to be a value of an additive set function P, which must satisfy the following three conditions: AXIOM 1. P(A)  0 for any event A in S. AXIOM 2. P(S) = 1

Axiomatic Approach AXIOM 3. If A1, A2 …, Ak is a finite collection of mutually exclusive events in S, then

Probability - Classical Approach If an experiment can result in n equally likely and mutually exclusive ways, and if nA of these outcomes have the characteristic A, then the probability of the occurrence of A, denoted by P(A) is defined to be the fraction

Frequency of Empirical Approach If an experiment is repeated or conducted n times, and if a particular attribute A occurred fA times, then an estimate of the probability of the event A is defined as: Note that Remark: Probability can be interpreted as relative frequency in the long run.

Example An experiment consists of tossing a fair coin three times in sequence. What is the probability that 2 heads will occur?

Example An biased coin (likelihood of a head is 0.75) is tossed three times in sequence. What is the probability that 2 heads will occur?

Relative Frequency vs. n 1 1 2 3 . . . n n = number of experiments performed

Probability - Subjective Approach Definition The probability P(A) is a measure of the degree of belief one holds in a specified proposition A. Note: Under this interpretation, probability may be directly related to the betting odds one would wager on the stated proposition. Odds The relative chances for the event A and the event that A does not occur, i.e., odds in favor of A