Introduction to Set Theory

Introduction to Set Theory - ( (...

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Introduction to Set Theory (and brief discussion of probability) Population the collection of all individuals or items under consideration In this course we denote population by Ω (the capital letter omega). Example: If we roll a die one time, then Ω ={1,2,3,4,5,6}. Ω represents all POSSIBLE OUTCOMES from the “random experiment” or the model under consideration. Loosely speaking, probability is a percentage. More specifically, we have the following concept: Suppose that a member is selected at random from a finite population. Then the probability that the member obtained has a specified attribute equals the percentage of the population that has that attribute. Example: Suppose a Stat 225 class consists of 30 males and 10 females. What is the probability that a randomly selected student [from this particular Stat 225 class] is male? Random Experiment an action whose outcome cannot be predicted with certainty (there is some randomness involved in the action). Event some SPECIFIED result that may or may not occur when the random experiment is performed. Remember, an event can be simple (only one specified result) or it can be a collection of results. (The class example of rolling a die: A 5 is rolled, B 3 is rolled, C an odd is rolled. A and B are simple events but C is not). Frequentist Interpretation of Probability The probability of an event is the long run proportion of times that the event occurs in independent repetitions of the random experiment. (Independence is a key topic in probability.) This can be written as: ( ) ( ) n E p E n = . The n represents the sample size. p(E) is the probability of event E occurring (p always represents probability; when clarification is not needed we will just use p). n(E) is
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