STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 1 1 Complement of A A not A A Ac contains all elements that are not in A Intersection of A and B A B A and B A B contains all elements that are in A and in B Union of A and B A B A or B contains all elements that are either in A or in B or both Axiom 1 Let A be any event defined over S Then P A 0 Axiom 2 P S 1 Axiom 3 If A 1 A 2 A 3 are events and A i A j for each i j then P A1 A2 Ak P A1 P A2 P Ak for each positive integer k and P A1 A2 A3 P A1 P A2 P A3 for an infinite but countable number of events Theorem 1 P A 1 P A Theorem 2 P 0 Theorem 3 If A B then P A P B Theorem 4 For any event A P A 1 For any event A 0 P A 1 P S 1 where S is the sample space Theorem 5 If A and B are any two events then P A B P A P B P A B P A or B P A P B P A and B Theorem 6 P A B C P A P B P C P A B P A C P B C P A B C P A B C D P A P B P C P D P A B P A C P A D P B C P B D P C D P A B C P A B D P A C D P B C D P A B C D 1 Suppose a 6 sided die is rolled The sample space S is Consider the following events 1 2 3 4 5 6 A the outcome is even B the outcome is greater than 3 a List outcomes in A B A A B A B b Find the probabilities P A P B P A P A B P A B if the die is balanced fair c Suppose the die is loaded so that the probability of an outcome is proportional to the outcome i e P 1 p P 2 2 p P 3 3 p P 4 4 p P 5 5 p P 6 6 p i Find the value of p that would make this a valid probability model ii Find the probabilities P A P B P A P A B P A B 2 Consider a thick coin with three possible outcomes of a toss Heads Tails and Edge for which Heads and Tails are equally likely but Heads is five times as likely than Edge What is the probability of Heads 3 The probability that a randomly selected student at Anytown College owns a bicycle is 0 55 the probability that a student owns a car is 0 30 and the probability that a student owns both is 0 10 a What is the probability that a student selected at random does not own a bicycle b What is the probability that a student selected at random owns either a car or a bicycle or both c What is the probability that a student selected at random has neither a car nor a bicycle C B B C 4 During the first week of the semester 80 of customers at a local convenience store bought either beer or potato chips or both 60 bought potato chips 30 of the customers bought both beer and potato chips What proportion of customers bought beer 5 Suppose P A 0 22 P B 0 25 P C 0 28 P A B 0 11 P A C 0 05 P B C 0 07 P A B C 0 01 Find the following a P A B b P A B c P A B C d P A B C e P A B C f P A B C g P A B C h P B C A 6 Let a 2 Suppose S 0 1 2 3 and P 0 c P k 1 ak k 1 2 3 a Find the value of c c will depend on a that makes this is a valid probability distribution b Find the probability of an odd outcome 7 Suppose S 0 1 2 3 and P 0 p Find the value of P k 1 2k k k 1 2 3 p that would make this a valid probability model
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