STAT 400 Lecture AL1 Answers for 1 4 Spring 2015 Dalpiaz Events A and B are independent if and only if P B A P B P A B P A P A B P A P B Note that if two events A and B are mutually exclusive then P A B 0 Therefore two mutually exclusive events cannot be independent unless at least one of them has probability 0 1 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 Are events a student owns a bicycle and a student owns a car independent P B C P B P C 0 10 0 55 0 30 B and C are NOT independent 1 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 Are events a randomly selected customer bought potato chips and a randomly selected customer bought beer independent Recall that P Beer 0 50 P B PC P B P PC 0 30 0 50 0 60 B and PC are independent Events A B and C are independent if and only if P A B P A P B P A C P A P C P B C P B P C and P A B C P A P B P C 1 Suppose that a fair coin is tossed twice Consider A H on the first toss B H on the second toss and C exactly one H in two tosses S TT TH HT HH A HT HH B TH HH C TH HT P A 1 2 P B 1 2 P C 1 2 a Are A and B independent A B HH P A B 1 4 P A B P A P B b P A C 1 4 P A C P A P C A and C are independent Are B and C independent B C TH P B C 1 4 P B C P B P C d A and B are independent Are A and C independent A C HT b B and C are independent Are A B and C independent Since P A B C P A P B P C A B and C are not independent even though A B and C are pairwise independent 2 A girl is told by her boyfriend that she is one in a billion She has a dimple in her chin probability 1 100 eyes of different colors probability 1 1 000 and is absolutely crazy about mathematics probability 1 10 000 a Do these events seem to be independent or dependent Independent b Show why the girl is one in a billion 1 3 100 1 1 000 1 10 000 1 1 000 000 000 1 1 billion Bart and Nelson talked Milhouse into throwing water balloons at Principal Skinner Suppose that Bart hits his target with probability 0 80 Nelson misses 25 of the time and Milhouse hits the target half the time Assume that their attempts are independent of each other P B 0 80 P N 0 75 P M 0 50 8 possible outcomes not equally likely a B N M B N M B N M B N M B N M B N M B N M B N M Find the probability that all of them will hit Principal Skinner P all P B N M P B P N P M 0 80 0 75 0 50 0 30 b Find the probability that exactly one of the boys will hit Principal Skinner P B N M P B P N P M 0 80 0 25 0 50 0 100 P B N M P B P N P M 0 20 0 75 0 50 0 075 P B N M P B P N P M 0 20 0 25 0 50 0 025 P exactly one 0 100 0 075 0 025 0 20 c Find the probability that at least one of the boys will hit Principal Skinner P at least one 1 P none 1 P B N M 1 P B P N P M 1 0 20 0 25 0 50 0 975 Idea P at least one of A i occurs 1 P none of A i occurs P A 1 or A 2 or or A n 1 P not A 1 and not A 2 and and not A n P A 1 A 2 A n 1 P A 1 A 2 A n For independent events P A 1 or A 2 or or A n 1 P not A 1 P not A 2 P not A n P A 1 A 2 A n 1 P A 1 P A 2 P A n 4 Often On time Parcel Service OOPS delivers a package to the wrong address with probability 0 05 on any delivery Suppose that each delivery is independent of all the others There were 7 packages delivered on a particular day What is the probability that at least one of them was delivered to the wrong address P at least one wrong 1 P all correct 1 0 95 7 0 3016627 5 A major oil company has decided to drill independent test wells in the Alaskan wilderness The probability of any well producing oil is 0 30 Find the probability that the fifth well is the first to produce oil F F F F S 0 70 0 70 0 70 0 70 0 30 0 07203 5 Consider a fair balanced symmetric 6 sided die Assuming that the outcome of each roll is independent of all other rolls find the probability that the first 6 occurs on the m th roll m th roll first m 1 rolls no 6 on the 1st roll no 6 on the m 1 th roll 6 on the m th roll 5 6 m 1 1 6 5 Consider a fair balanced symmetric 6 sided die Assuming that the outcome of each roll is independent of all other rolls find the probability that 6 occurs before an odd outcome 1 2 1 2 P 6 before odd 6 6 6 6 2 1 2 6 6 1 1 1 0 25 4 6 1 2 6 k k 1 2 1 6 k 0 6 6 2 6 P not 6 and not odd OR Four relevant outcomes P 6 before odd 1 4 0 25 one good outcome 6 three bad outcomes 1 3 5 6 An automobile salesman thinks that the probability of making a sale is 0 30 If he talks to five customers on a particular day what is the probability that he will make exactly 2 sales Assume independence S S S S F S F F F S F S F F S …
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