Lecture 3.pdf - Lecture3:Probability ,youshouldbeableto...

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2/22/2017 1 After completing this lecture, you should be able to: Explain basic probability concepts and definitions Use a Venn diagram or tree diagram to illustrate simple probabilities Apply common rules of probability Compute conditional probabilities Determine whether events are statistically independent Use Bayes’ Theorem for conditional probabilities Lecture 3: Probability Important Terms Random Experiment – is a process leading to two or more possible (an uncertain) outcomes, without knowing exactly which outcome will occur. Basic Outcome – is a possible outcome of a random experiment. Sample Space ( ) – is the set or collection of all basic outcomes of a random experiment. Event ( ) – is any subset of basic outcomes from the sample space. Intersection of Events – If ܣ and ܤ are two events in a sample space S, then the intersection, ܣ ∩ ܤ , is the set of all outcomes in ܵ that belong to both and (joint probability of ܣ and ܤ for the intersection of ܣ and ܤ ). ܣ ܤ ܣ ∩ ܤ ܵ Important Terms ܣ and ܤ are Mutually Exclusive Events if they have no basic outcomes in common i.e., the set ܣ ∩ ܤ is empty ܣ ܤ ܵ Union of Events – If ܣ and ܤ are two events in a sample space ܵ , then the union, ܣ ∪ ܤ , is the set of all outcomes in ܵ that belong to either or The entire shaded area represents ܣ ∪ ܤ ܣ ܤ ܵ
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2/22/2017 2 Important Terms Events ܧ , ܧ , … , ܧ are Collectively Exhaustive events if ܧ ∪ ܧ ∪ ⋯ ∪ ܧ ൌ ܵ i.e., the events completely cover the sample space The Complement of an event ܣ is the set of all basic outcomes in the sample space that do not belong to ܣ . The complement is denoted ܣ ̅ ܣ ܵ ܣ̅ Examples Let the Sample Space be the collection of all possible outcomes of rolling one die: ܵ ൌ ሾ1, 2, 3, 4, 5, 6ሿ Let ܣ be the event “Number rolled is even” Let ܤ be the event “Number rolled is at least 4” Then ܣ ൌ ሾ2, 4, 6ሿ and ܤ ൌ ሾ4, 5, 6ሿ Complements: ܣ ̅ ൌ ሾ1, 3, 5ሿ ܤ ൌ ሾ1, 2, 3ሿ Intersections: ܣ ∩ ܤ ൌ ሾ4, 6ሿ ܣ ̅ ∩ ܤ ൌ [5] Unions: ܣ ∪ ܤ ൌ ሾ2, 4, 5, 6ሿ ܣ ∪ ܣ ̅ 1, 2, 3, 4, 5, 6 ൌ ܵ ܵ ൌ ሾ1, 2, 3, 4, 5, 6ሿ ܣ ൌ ሾ2, 4, 6ሿ ܤ ൌ ሾ4, 5, 6ሿ Examples Mutually exclusive: ܣ and ܤ are not mutually exclusive, i.e., The outcomes 4 and 6 are common to both Collectively exhaustive: A and B are not collectively exhaustive, i.e., ܣ ∪ ܤ does not contain 1 or 3
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2/22/2017 3 Probability and Its Postulates Probability is the chance that an uncertain event will occur (always between 0 and 1) 0 ൑ ܲሺܣሻ ൑ 1 For any event ܣ Certain Impossible .5 1 0 There are three definitions for the probability of an uncertain event: 1. classical probability 2. relative frequency probability 3. subjective probability Classical Probability Classical probability is the proportion of times that an event will occur, assuming that all outcomes in a sample space are equally likely to occur .
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