1 Probability- Basic Concepts and Approaches Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems Stracener_EMIS 7370/STAT 5340_Sum 08_

Stracener_EMIS 7370/STAT 5340_Sum 08_ Probability-Basic Concepts and Approaches Basic Terminology & Notation Basic Concepts Approaches to Probability Axiomatic Classical (A Priori) Frequency or Empirical (A Posteriori) Subjective

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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. Basic Terminology

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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. Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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 . Basic Terminology

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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´A´ S Venn Diagram Operations With Events

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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 Basic Concept

Stracener_EMIS 7370/STAT 5340_Sum 08_ (1) First, there is a question of what we mean when we say that a probability is 0.82, or 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 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? Probability-Basic Questions

Stracener_EMIS 7370/STAT 5340_Sum 08_ Axiomatic Classical (A Priori) Frequency or Empirical (A Posteriori) Subjective Approaches to Probability

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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

Stracener_EMIS 7370/STAT 5340_Sum 08_ AXIOM 3. If A 1, A 2 …, A k is a finite collection of mutually exclusive events in S, then Axiomatic Approach

Stracener_EMIS 7370/STAT 5340_Sum 08_ If an experiment can result in n equally likely and mutually exclusive ways, and if n A 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 Probability - Classical Approach

Stracener_EMIS 7370/STAT 5340_Sum 08_ If an experiment is repeated or conducted n times, and if a particular attribute A occurred f A 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. Frequency of Empirical Approach

Stracener_EMIS 7370/STAT 5340_Sum 08_ An experiment consists of tossing a fair coin three times in sequence. What is the probability that 2 heads will occur? Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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? Example

Stracener_EMIS 7370/STAT 5340_Sum 08_ Relative Frequency vs. n n = number of experiments performed n 0 1

Stracener_EMIS 7370/STAT 5340_Sum 08_ 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 Probability - Subjective Approach