STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Answers for 1 5 In Neverland men constitute 60 of the labor force The rates of unemployment are 6 0 and 4 5 among males and females respectively A person is selected at random from Neverland s labor force P U M 0 06 P M 0 60 a P U F 0 045 What is the probability that the person selected is a male and is unemployed P M U P M P U M 0 60 0 06 0 036 b What is the probability that the person selected is a female and is unemployed P F U P F P U F 0 40 0 045 0 018 Unemployed Employed Total Male 0 036 0 564 0 60 Female 0 018 0 382 0 40 0 054 0 946 1 00 Total c What is the probability that the person selected is unemployed P U 0 036 0 018 0 054 OR Law of Total Probability P U P M P U M P F P U F 0 60 0 06 0 40 0 045 0 054 d Suppose the person selected is unemployed What is the probability that a male was selected P M U 0 036 0 054 2 3 OR Bayes Theorem P M U 0 60 0 06 P M P U M 2 0 60 0 06 0 40 0 045 3 P M P U M P F P U F 2 In a presidential race in Neverland the incumbent Democrat D is running against a field of four Republicans R 1 R 2 R 3 R 4 seeking the nomination Political pundits estimate that the probabilities of R 1 R 2 R 3 and R 4 winning the nomination are 0 40 0 30 0 20 and 0 10 respectively Furthermore results from a variety of polls are suggesting that D would have a 55 chance of defeating R 1 in the general election a 70 chance of defeating R 2 a 60 chance of defeating R 3 and an 80 chance of defeating R 4 Assuming all these estimates to be accurate what are the chances that D will be a two term president P R 1 0 40 P R 2 0 30 P R 3 0 20 P R 4 0 10 P W R 1 0 55 P W R 2 0 70 P W R 3 0 60 P W R 4 0 80 Law of Total Probability P W P W R 1 P W R 2 P W R 3 P W R 4 P R 1 P W R 1 P R 2 P W R 2 P R 3 P W R 3 P R 4 P W R 4 0 40 0 55 0 30 0 70 0 20 0 60 0 10 0 80 0 63 W L R1 R2 R3 R4 0 40 0 55 0 30 0 70 0 20 0 60 0 10 0 80 0 22 0 21 0 12 0 08 0 18 0 09 0 08 0 02 0 37 0 40 0 30 0 20 0 10 1 00 0 63 3 In Anytown 10 of the people leave their keys in the ignition of their cars Anytown s police records indicate that 4 2 of the cars with keys left in the ignition are stolen On the other hand only 0 2 of the cars without keys left in the ignition are stolen Suppose a car in Anytown is stolen What is the probability that the keys were left in the ignition P Keys 0 10 P Keys 1 0 10 0 90 P Stolen Keys 0 042 P Stolen Keys 0 002 Need P Keys Stolen Bayes Theorem P Keys Stolen P Keys P Stolen Keys P Keys P Stolen Keys P Keys P Stolen Keys 0 10 0 042 0 70 0 10 0 042 0 90 0 002 OR Stolen Stolen 0 042 0 10 Keys 0 0042 0 0958 0 10 0 0018 0 8982 0 90 0 0060 0 9940 1 00 0 002 0 90 Keys P Keys Stolen P Keys Stolen 0 0042 0 70 P Stolen 0 0060 OR P Stolen 0 0042 0 0018 0 0060 P Keys Stolen P Keys Stolen 0 0042 0 70 P Stolen 0 0060 3 A warehouse receives widgets from three different manufacturers A 50 B 30 and C 20 Suppose that 2 of the widgets coming from A are defective as are 4 of the widgets coming from B and 7 coming from C a Find the probability that a widget selected at random at this warehouse is defective Law of Total Probability P D P A P D A P B P D B P C P D C 0 50 0 02 0 30 0 04 0 20 0 07 0 010 0 012 0 014 0 036 b Suppose a widget that was selected at random is found to be defective What is the probability that it came from manufacturer A Manufacturer B Manufacturer C P A D 0 010 5 0 036 18 P B D 0 012 6 0 036 18 P C D 7 0 014 0 036 18 3 Seventy percent of the light aircraft that disappear while in flight in Neverland are subsequently discovered Of the aircraft that are discovered 60 have an emergency locator whereas 90 of the aircraft not discovered do not have such a locator Suppose a light aircraft that has just disappeared has an emergency locator What is the probability that it will not be discovered P Discovered 0 70 P Discovered 1 0 70 0 30 P Locator Discovered 0 60 P Locator Discovered 0 90 Need P Discovered Locator Locator Locator 0 60 0 70 Discovered 0 42 0 28 0 70 0 90 0 30 Discovered 0 03 0 27 0 30 0 45 0 55 1 00 P Discovered Locator P Discovered Locator 0 03 1 0 06667 P Locator 0 45 15 4 In a certain population the proportion of individuals who have a particular disease is 0 025 A test for the disease is positive in 94 of the people who have the disease and in 4 of the people who do not a P D 0 04 P D 0 94 P D 0 025 Find the probability of receiving a positive reaction from this test Need P 0 025 0 94 D 0 0235 0 975 0 04 D 0 0390 0 0625 0 0015 0 025 0 9360 0 975 0 9375 1 000 OR Law of Total Probability P P D P D P D P D 0 025 0 94 0 975 0 04 0 0625 b If a person received a positive reaction from this test what is the probability that he she has the disease P D 0 0235 0 0625 0 376 OR Bayes Theorem P D c 0 025 0 94 0 376 0 025 0 94 0 975 0 04 If a person received a negative reaction from …
View Full Document
Unlocking...