assignment 1 - monash university · assignment 1 1 efron’s intransitive dices a) efron’s...

Assignment 1 1 Efron’s Intransitive Dices a) Efron’s intransitive dices (see figure above) are as follow: A: 4, 4, 4, 4, 0, 0 B: 3, 3, 3, 3, 3, 3 C: 6, 6, 2, 2, 2, 2 D: 5, 5, 5, 1, 1, 1 Compute the mean (expectation) and variance for each of these four dices and a normal dice. (2 marks) b) Lets assume we play, that the dice with the larger value wins. You are allowed to pick one of the four dices first and I will pick one of the three remaining. We than throw our dice and the larger values wins. Who will, in statistical average, win? Me (who picks last) or you who picks the dice first? Explain why. Note, this question is not about which dice is in average the best! (2 marks) 2 Probabilities A student of Monash needs in average 23 min. for the daily trip to the lectures, with a standard deviation of 3 min. We assume that the time needed for the trip is normal distributed (is that actually possible?). a) If the aforesaid student leaves his dormitory (home) well rested at 9:05 am and the lecture starts at 9:30 am, how often (in percent) will the student reach the lecture in time and how often will the student be late by more than 5 minutes? (3 marks) b) Of cause the good student does not want to be late. What is the probability that the student is late at least once during the semester term (12 lectures)? (1 mark) c) When should the good student leave his dormitory (home), if the probability of being in time for all lectures of the semester is 90%? (2 marks) Hint: You can use the following approximation of the standarized normal cumulative distribution: 3a Central Limit Theorem (you can either do 3a or 3b, as you like) The standard deviation of the daily mean temperature time series is σ24hrs = 2.78K and that of the monthly mean time series is σmon = 1.37K. Try, with the help of the central limit theorem, to estimate the number of degree of freedom per month. What can you learn from this value? (2 marks) 3b Base Rate Neglect (you can either do 3a or 3b, as you like) Lets assume the theoretical probability distribution of SETU (students evaluation of the unit) marks (values 1-5) for this unit is [0% 0% 20% 50% 30%] for the value 1 to 5. So the theoretical average is 4.1. If you have just one student mark, than you have a 30% chance to get a total SETU average mark of 5.0. Now how likely is it that you get a score total SETU average of 5.0 as function of student numbers? Test it with student number of 5, 10, 50 and 500. What does this mean about comparing total SETU average based on different student numbers (base rates)? (2 marks)

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Assignment 1

1 Efron’s Intransitive Dices

a)Efron’sintransitivedices(seefigureabove)areasfollow:A:4,4,4,4,0,0B:3,3,3,3,3,3C:6,6,2,2,2,2D:5,5,5,1,1,1Computethemean(expectation)andvarianceforeachofthesefourdicesandanormaldice.(2marks)b)Letsassumeweplay,thatthedicewiththelargervaluewins.YouareallowedtopickoneofthefourdicesfirstandIwillpickoneofthethreeremaining.Wethanthrowourdiceandthelargervalueswins.Whowill,instatisticalaverage,win?Me(whopickslast)oryouwhopicksthedice first? Explainwhy.Note, thisquestion isnotaboutwhichdice is inaveragethebest!(2marks)

2 Probabilities A student of Monash needs in average 23 min. for the daily trip to the lectures, with astandard deviation of 3 min. We assume that the time needed for the trip is normaldistributed(isthatactuallypossible?).

a) If the aforesaid student leaves his dormitory (home) well rested at 9:05 am and thelecturestartsat9:30am,howoften(inpercent)will thestudentreachthe lecture intimeandhowoftenwillthestudentbelatebymorethan5minutes?(3marks)

b) Of cause the good student does not want to be late. What is the probability that thestudentislateatleastonceduringthesemesterterm(12lectures)?(1mark)

c)Whenshouldthegoodstudentleavehisdormitory(home),iftheprobabilityofbeingintimeforalllecturesofthesemesteris90%?(2marks)

Hint: You can use the following approximation of the standarized normal cumulativedistribution:

3a Central Limit Theorem (you can either do 3a or 3b, as you like) The standard deviation of the daily mean temperature time series is σ24hrs = 2.78K and that of the monthly mean time series is σmon = 1.37K. Try, with the help of the central limit theorem, to estimate the number of degree of freedom per month. What can you learn from this value? (2marks)

3b Base Rate Neglect (you can either do 3a or 3b, as you like) Lets assume the theoretical probability distribution of SETU (students evaluation of the unit) marks (values 1-5) for this unit is [0% 0% 20% 50% 30%] for the value 1 to 5. So the theoretical average is 4.1. If you have just one student mark, than you have a 30% chance to get a total SETU average mark of 5.0. Now how likely is it that you get a score total SETU average of 5.0 as function of student numbers? Test it with student number of 5, 10, 50 and 500. What does this mean about comparing total SETU average based on different student numbers (base rates)? (2marks)