Presentation on theme: "Basic Concepts and Approaches"— Presentation transcript:
1 Basic Concepts and Approaches EMIS 7370 STAT 5340Department of Engineering Management, Information and SystemsProbability and Statistics for Scientists and EngineersProbability-Basic Concepts and ApproachesDr. Jerrell T. Stracener, SAE FellowLeadership in Engineering
2 Probability-Basic Concepts and Approaches Basic Terminology & NotationBasic ConceptsApproaches to ProbabilityAxiomaticClassical (A Priori)Frequency or Empirical (A Posteriori)Subjective
3 Basic Terminology Any well-defined action. It is any action or process Definition – ExperimentAny well-defined action. It is any action or processthat generates observations.Definition - OutcomeThe result of performing an experiment
4 Basic Terminology Definition - Sample Space The set of all possible outcomes of a statistical experiment iscalled the sample space and is represented by S.Remark: Each outcome in a sample space is called anelement or a member of the sample space or simply a samplepoint.
5 ExampleAn 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.
6 ExampleAn 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.
7 Basic Terminology Definition - Event An event is the set of outcomes of the sample space each havinga given characteristic or attributeRemark: An event, A, is a subset of a sample space, S, i.e.,A S.
8 Basic Terminology Definition - Types of Events If an event is a set containing only one element or outcomeof the sample space, then it is called a simple event. Acompound event is one that can be expressed as the unionof simple events.Definition - Null EventThe null event or empty space is a subset of the sample spacethat contains no elements. We denote the event by the symbol.
9 Operations With Events Certain operations with events will result in the formation of newevents. These new events will be subsets of the same samplespace as the given events.Definition - The intersection of two events A and B, denoted bythe symbol A B, or by AB is the event containing all elementsthat are common to A and B.Definition - Two events A and B are mutually exclusive ifA B = .Definition - The union of two events A and B, denoted by thesymbol A B, is the event containing all the elements thatbelong to A or to B or to both.
10 Operations With Events Definition - The complement of an event A with respect to Sis the set of all elements of S that are not in A. We denote thecomplement 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.AA´SVenn Diagram
11 Basic Concept For any event A in S, the probability of A occurring is a number between 0 and 1, inclusive, i.e.,whereandwhere Ø is the null event
12 Probability-Basic Questions (1) First, there is a question of what we mean when we saythat a probability is 0.82, or 0.25.- What is probability?(2) Then, there is the question of how to obtain numericalvalues of probabilities, i.e., how do we determine thata certain probability is 0.82, or 0.25.- How is probability determined?(3) Finally, there is the question of how probabilities canbe combined to obtain other probabilities.- What are the rules of probability?
13 Approaches to Probability AxiomaticClassical (A Priori)Frequency or Empirical (A Posteriori)Subjective
14 Axiomatic ApproachGiven a finite sample space S and an event A in S, wedefine P(A), the probability of A, to be a value of an additive setfunction P, which must satisfy the following three conditions:AXIOM 1.P(A) for any event A in S.AXIOM 2.P(S) = 1
15 Axiomatic Approach AXIOM 3. If A1, A2 …, Ak is a finite collection of mutuallyexclusive events in S, then
16 Probability - Classical Approach If an experiment can result in n equally likely and mutuallyexclusive ways, and if nA of these outcomes have thecharacteristic A, then the probability of the occurrence of A,denoted by P(A) is defined to be the fraction
17 Frequency of Empirical Approach If an experiment is repeated or conducted n times, andif a particular attribute A occurred fA times, then an estimate ofthe probability of the event A is defined as:Note thatRemark: Probability can be interpreted as relative frequency inthe long run.
18 ExampleAn experiment consists of tossing a fair coin three times in sequence.What is the probability that 2 heads will occur?
19 ExampleAn biased coin (likelihood of a head is 0.75) is tossed three times in sequence.What is the probability that 2 heads will occur?
20 Relative Frequency vs. n 1nn = number of experiments performed
21 Probability - Subjective Approach DefinitionThe probability P(A) is a measure of the degree of beliefone holds in a specified proposition A.Note: Under this interpretation, probability may be directly relatedto the betting odds one would wager on the stated proposition.OddsThe relative chances for the event A and the event that Adoes not occur, i.e.,odds in favor of A