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YMS Chapter 6 Probability: Foundations for Inference 6.1 – The Idea of Probability
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Probability – 3 Interpretations ► Any outcome of any random phenomenon is the proportion of times it would occur in a very long series of repetitions ► Long-term relative frequency ► Branch of math that describes the pattern of chance outcomes
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► When individual outcomes are uncertain but there is still a regular distribution of outcomes in the long run ► Relative frequencies of outcomes seem to settle down to fixed values in the long run ► Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run Randomness – 3 Interpretations
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Exploring Randomness ► Must have a long series of independent trials ► Probability is empirical (based on previous experience) ► Computer simulations are very useful 6.1 Practice – p334 #6.4, 6.9 and 6.10
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YMS 6.2 Probability Models
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► Sample space (S) The set of all possible outcomes ► Event Any outcome or set of outcomes of a random phenomenon A subset of S ► Probability model A mathematical description of a random phenomenon consisting of two parts: S and the assignment of probabilities to events ► Multiplication (Counting) Principle Multiply number of outcomes for each event to find total number of ways Use a tree diagram to visually represent and find sample space
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► Any probability is a number between 0 and 1. 0 < P(A) < 1 ► All possible outcomes together must have probability 1. P(S) = 1 ► Complement Rule - The probability that an event does not occur is 1 minus the probability that it does. P(A C ) = 1- P(A) Probability Rules
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► Disjoint or Mutually Exclusive Events If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. P(A or B) = P(A) + P(B) – P(A and B) ► Independent events Knowing that if one event occurs it does not change the probability that the other occurs P(A and B) = P(A)P(B)
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Basic Set Theory ► Union Combination of elements in sets ► Intersection What the sets have in common ► Null set Set without elements ► Venn Diagrams Very useful to create when answering questions about relationships among sets 6.2 Practice – p340 #6.15, 6.19, 6.26, 6.28, 6.29, 6.35, 6.42
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YMS 6.3 General Probability Rules
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► Rule for disjoint events P(one or more of A, B, C) = P(A) + P(B) + P(C) ► Two events are independent if P(B|A)=P(B) ► Conditional Probability The probability of one event under the condition that we know another event
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► General Multiplication Rule (rewrite for conditional probability) P(A and B) = P(A)*P(B|A) ► Bayes’s Rule Don’t memorize! Use Tree Diagrams 6.3 Practice – p365 #6.51, 6.52, 6.58, 6.59, 6.61, 6.64, 6.65
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