Chp_1 - Probability Sample Spaces and Events Probability...

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Probability Sample Spaces and Events Probability The Axioms of Probability Some Elementary Theorems Conditional probability Bayes’ Theorem
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Sample Space and Events A set of all possible outcomes of an experiment is called a sample space . It is denoted by S . A particular outcome, i.e. an element in S is called a sample point . An experiment consists of the simple process of noting outcomes. The outcome of an experiment may be a simple choice between two alternatives; it may be result of a direct measurement or count; or it may be answer obtained after extensive measurements and calculations.
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Sample Space and Events (cont’d) A sample space is said to be discrete if it has finitely many or a countable (denumerable) infinity of elements. If the elements of a sample space constitute a continuum, the sample space is said to be continuous . For example, all points on a line, all points on a line segment, or all points in a plane. Any subset of a sample space is called an event. The subset { a } consisting of a single sample point a S is called an elementary event. 2200 φ and S itself are subsets of S and so are events.
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Sample Space and Events (cont’d) Example 1: A technician has to check the suitability of 3 solid crystal lasers and 2 CO 2 lasers for a given task. Let ( a,b ) represent the event that the technician will find a of the solid crystal lasers and b of the CO 2 lasers suitable for the task, then the sample space will be S = {(0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2), (3,0), (3,1), (3,2)}.
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Sample Space and Events (cont’d) R = {(0,0), (1,1), (2,2)} is the event that equally many solid crystal lasers and CO 2 lasers are suitable. T = {(0,0), (1,0), (2,0), (3,0)} is the event that none of the CO 2 lasers are suitable for the task. U = {(0,1), (0,2), (1,2)} is the event that fewer solid crystal laser than the CO 2 lasers are suitable for the task.
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Sample Space and Events (cont’d) If two events have no element in common, they are called mutually exclusive events , i.e. they cannot occur simultaneously. In above example R T are not mutually exclusive events while R & U and T & U are mutually exclusive events. In order to discuss the basic concepts of the probabilistic model which we wish to develop, it will be very convenient to have available some ideas and concepts of mathematical theory of sets. Sample spaces and events, particularly relationship among events are depicted by means of Venn diagrams
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Sample Space and Events (cont’d) If A and B are any two sets in a sample space S , A B = { x S | x A or x B (or both)}; A B = { x S | x A and x B }; The complement of A i.e. A
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Chp_1 - Probability Sample Spaces and Events Probability...

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