Probability - Vineet Goyal Permutations Combinations Sample...

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QSRP - Chapter 8 1 Vineet Goyal Permutations Combinations Sample Spaces & Events Probability Conditional Probability Stochastic Processes Independent events Bayes’s Formula Quantitative Skills Review Program Introduction to Probability Chapter 8
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QSRP - Chapter 8 2 Vineet Goyal Permutations Combinations Sample Spaces & Events Probability Conditional Probability Stochastic Processes Independent events Bayes’s Formula Probability Definition Degree of likelihood that some future event will have a particular outcome Example Weather Coin Toss Performance of stocks
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QSRP - Chapter 8 3 Vineet Goyal Permutations Combinations Sample Spaces & Events Probability Conditional Probability Stochastic Processes Independent events Bayes’s Formula Sets - Definition and Notation The study of sets is a fundamental requirement for studying probability Definitions A Set is collection of objects these objects are called Members or Elements of the Set Notation Sets are denoted by upper case letters Elements are denoted by lower case letters If an element a belongs to a set X , we write a X “a belongs to X” If an element b does not belong to a set X , we write b X “b does not belong to X”
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QSRP - Chapter 8 4 Vineet Goyal Permutations Combinations Sample Spaces & Events Probability Conditional Probability Stochastic Processes Independent events Bayes’s Formula Sets - Construction/Description There are two ways of describing a set Roster Method: listing all members of the set Set of possible results of tossing a die = {1, 2, 3, 4, 5, 6} Set of vowels = {a, e, i, o, u} Property Method: Describing a property held by all members and no non- members Set of possible results of tossing a die = {x| x is an integer, 1<= x <= 6} Set of positive even integers = {x| x is an integer, x > 0, x is divisible by 2} Two ways of constructing sets construct the following using the property method {Amstel, Budweiser, Miller, Coors} construct the following using the roster method {x| x is a Prime Number}
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QSRP - Chapter 8 5 Vineet Goyal Permutations Combinations Sample Spaces & Events Probability Conditional Probability Stochastic Processes Independent events Bayes’s Formula Subsets - Definition and Notation Definition If each element of a set A belongs to a set B , then we call A a Subset of B A = {1, 2, 3} B = {1, 2, 3, 4, 5} Notation If A is a subset of B , we write A B “A is contained in B” “A is a subset of B” or B A “B contains A” “B is a superset of A”
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QSRP - Chapter 8 6 Vineet Goyal Permutations Combinations Sample Spaces & Events Probability Conditional Probability Stochastic Processes Independent events Bayes’s Formula Sets & Subsets - Some Results Equality and Inequality of Sets A and B are equal if they have exactly the same elements and we write A = B “A equals B” If A and B are not equal, then we write A B “A does not equal B” Obviousman says “for all sets A we have A A If we have A B and B A , then clearly A = B !
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