STAT 400 Lecture AL1 1 Answers for 1 1 Suppose a 6 sided die is rolled The sample space S is Spring 2015 Dalpiaz 1 2 3 4 5 6 Consider the following events A the outcome is even B the outcome is greater than 3 a List outcomes in A B A A B A B A the outcome is even 2 4 6 B the outcome is greater than 3 4 5 6 A 1 3 5 A B 4 6 A B 2 4 5 6 b Find the probabilities P A P B P A P A B P A B if the die is balanced fair P A 3 6 P B 3 6 P A 3 6 P A B 2 6 P A B 4 6 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 P 1 P 2 P 3 P 4 P 5 P 6 1 ii p 1 21 p 2 p 3 p 4 p 5 p 6 p 21 p 1 Find the probabilities P A P B P A P A B P A B P A P 2 P 4 P 6 2 21 4 21 6 21 12 21 P B P 4 P 5 P 6 4 21 5 21 6 21 15 21 P A 1 P A 1 12 21 9 21 OR P A P 1 P 3 P 5 1 21 3 21 5 21 9 21 P A B P 4 P 6 4 21 6 21 10 21 P A B P 2 P 4 P 5 P 6 2 21 4 21 5 21 6 21 17 21 OR P A B P A P B P A B 12 21 15 21 10 21 17 21 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 P Heads P Tails p for some p P Edge 1 p 5 P Heads P Tails P Edge 1 p p 1 11 p 1 5 P Heads p 3 5 11 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 P B 0 55 a p 1 5 P C 0 30 What is the probability that a student selected at random does not own a bicycle P B 1 P B 1 0 55 0 45 P B C 0 10 C C B 0 10 0 45 0 55 B 0 20 0 25 0 45 0 30 0 70 1 00 b What is the probability that a student selected at random owns either a car or a bicycle or both P B C P B P C P B C 0 55 0 30 0 10 0 75 OR P B C P B C P B C P B C 0 10 0 20 0 45 0 75 OR P B C 1 P B C 1 0 25 0 75 c What is the probability that a student selected at random has neither a car nor a bicycle P B C 0 25 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 P B PC 0 80 P PC 0 60 P B PC 0 30 P B PC P B P PC P B PC 0 80 P B 0 60 0 30 P B 0 50 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 a P A B 0 36 b P A B 0 64 c P A B C 0 53 d P A B C 0 47 e P A B C 0 17 f P A B C 0 75 g P A B C 0 11 h P B C A 0 88 6 Let a 2 Suppose S 0 1 2 3 and P 0 c a P k 1 ak k 1 2 3 Find the value of c c will depend on a that makes this is a valid probability distribution Must have p x 1 1 1 k 1 a k c all x bk k 0 1 1 b b 1 1 1 1 1 k k 1 1 1 1 a 1 k 1 a k 0 a a OR 1 1 1 1 1 1 k 1 a k a k 0 a k a 1 1 a a 1 c b 1 a 1 c 1 1 a 2 a 2 a 1 a 1 a 1 1 Find the probability of an odd outcome P odd p 1 p 3 p 5 1 a1 1 first term 1 base a 1 1 a 2 a 2 a 1 1 a3 1 a5 7 Suppose S 0 1 2 3 and P 0 p Find the value of Must have P k Since k 0 2k k k 1 2 3 p that would make this a valid probability model p x 1 all x 1 ak ea k 1 1 k k 1 2 k p 1 1 k 1 k k 1 2 k k 0 2 k Therefore p e 1 2 1 1 and p 2 e 1 2 e 1 2 1
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