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L2 Probability Numbers - Probability Numbers For many...

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Probability Numbers For many statisticians the concept of “the probability that an event occurs” is ultimately rooted in the interpretation of an event as an outcome of an experiment, others would interpret the concept as a subjective degree of belief that the event occurs. This difference in interpretation has been the source of great debate between Frequentists (the former group) and Bayesians (the latter group) within the statistics profession. This philosophical debate ultimately influences the way that empirical results are obtained and interpreted, but does not affect the use of probability ideas employed in describing the distribution of happiness in a society. Hence we shall proceed under the Frequentist interpretation whilst acknowledging the existing of an alternative. At the heart of the frequentists view of probability is the notion of an experiment. A sufficiently general definition of the term “Experiment” permits an almost universal application of the probability concept. Essentially a procedure must possess two properties to be eligible as an experiment. 1) It should be notionally repeatable an infinite number of times with a well defined common set of possible outcomes each time the procedure takes place. 2) There should be uncertainty as to which outcome will occur before the procedure takes place. Probability Theory is best understood by using ideas from set theory in its description. The appendix provides an outline of the basic set theory ideas that are used. The set of mutually exclusive (having nothing in common) and exhaustive (a complete list of) possible Basic Outcomes (denoted o i ) of an experiment is usually called The Sample Space (denoted S). An Event is defined as any subset of this sample space, including the empty set and the sample space itself, generally an event is denoted by an upper case letter say A, A c , the complement of A, then corresponds to A not happening. An event is said to have occurred when any one of the basic outcomes in its defining subset is realized. Thus, on executing the procedure, the empty or null set is the event “no outcome occurs” which is certain not to happen, similarly the sample space set (the universal set in set theoretic terms) is the event “an outcome occurs” which is certain to happen. Given outcome uncertainty there is obviously event uncertainty ranging in degree between the empty set and the sample space. Numbers attached to events reflecting the degree of certainty with which they occur, and which in turn obey certain coherency axioms, are called “Probabilities.” The coherency axioms are simple yet remarkably powerful, they provide the basis for all probability theory regardless of its objective or subjective foundations. Denoting the occurrence of the i’th basic outcomes as o i and the probability of it happening as P(o i ) and denoting
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events by upper case letters (with S and respectively reserved for the sample and empty sets) the coherency axioms are as follows: 1)
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