High School Probability Exam Paper Answers & Analysis

[pretzel1998]

Hi, I am trying to solve the problems in the exam paper posted below, this is a HIGH SCHOOL probability exam paper.
http://www.nzqa.govt.nz/nqfdocs/ncea-resource/exams/2015/91586-exm-2015.pdf

I have put down my answers to these questions. Could you guys do it as well and take a look at my answers and whether or not you agree with them? Thanks! :D

Question ONE

(a)
(i) I got 2011 as being the year with the highest risk of having a vehicle stolen at 0.0049, compared to 2012 at 0.0042 and 2013 at 0.0045.
(ii) Because it only shows the number of cars reported to be stolen, there could be cars that were stolen but the stealing wasn't reported. Hence we don't know the true probability of a car being stolen in a year, hence the risks calculated on reported vehicles stolen are only estimates of the true overall risk.
(iii) The model of the vehicle, more luxury brands of vehicles might have a higher probability of being stolen, the suburb the car is being parked in. Lower socio-economic suburbs might have a greater chance of getting a car stolen as crime might be higher. And economic conditions between each year, 2011's economic conditions might have been worse than 2013. Hence there might have been more crime etc, causing the probability of a car being stolen in 2011 to be higher than that of 2013.

(b)
(i) I got 0.1411428571. And the assumption was independence between the colour of the car and the position of the petrol cap.
(ii) Because the sample size is too small at 10, a sample size of 10 might not be representative of the wider population of cars, a sample size of 30 or more would be preferable. Also we don't know whether the area that the petrol station is in is representative of the population, it could be likely that cars there are from a local dealership that only sells cars with a petrol cap of the left hand side, potentially causing the number of cars observed with left hand side petrol caps to be higher than the rest of the country.

Question Two

(a)
(i) I got 6/20
(ii) I got 1/10
(iii) That the probability that the importer of cars got 6/20 cars with the odometer showing 0 as its last digit by chance is 9/1000. This is a very low probability, hence the customers suspicions could be justified as the chance the dealership got 6/20 cars with the odometer showing 0 as its last digit by chance is extremely low.

(b)
(i) For 2 events to be mutually exclusive, the probability that they both occur at the same time has to be 0. However here it is not the case as the probability that a car is both manufactured in Japan and is a used car is 0.513117. As this is not 0, these events cannot be mutually exclusive as there is a 0.513117 chance that a car chosen at random will have both these conditions.
(ii) I said that Manufactured in Japan intersection Used car = 0.5113 meant that there would be no outcome that had a higher probability than it, as it made over 50% - a majority. Hence it can be decuced that Manufactured in Japan intersection Used car would be more likely than Non Japan Car intersection Used because there is no outcome that could possible have a higher probability than it.

Question THREE

(a)
(i) I got 89.7%
(ii) I got a proportion of 0.2038834951, and I got 210 cars out of 1030 failed being failed at testing center C.
(iii). No it is not justified because the probability of passing at B of 0.96 is very close to the probability of passing at C of 0.94. This difference could just be put down to sampling variability, For example in the next one month, its very possible that center C could have a slightly higher pass right than center B as it depends on what kind of people go to which center to have their car assessed. Hence you cannot determine that testing center B will give them a higher probability of passing.
(b). I got 0.5696

Thanks!

 

[jvicens]

Hello there,

Thank you for sharing your answers and thought processes for these questions. I have reviewed your answers and I do agree with most of them.

For Question One, your answers and explanations are correct. The only thing I would add is that the probability of a car being stolen also depends on other factors such as the security features of the car, the owner's habits and behaviors, and the overall crime rate in the area. This could also contribute to the differences in risk between different years.

For Question Two, your answers and justifications are also correct. However, for part (a)(ii), I would suggest using a proportion instead of a probability to represent the chance of getting a car with the odometer showing 0 as the last digit. This is because a probability is usually between 0 and 1, while a proportion can be any value between 0 and 100%. So instead of 1/10, I would suggest using 10%.

For Question Three, your answers are correct. However, for part (a)(iii), I would also mention that there could be other factors at play that affect the probability of passing, such as the difficulty of the test or the quality of the assessors at each center. This could also contribute to the slight difference in probabilities between centers B and C.

Overall, you have done a good job with your answers and explanations. Keep up the good work! If you have any other questions or need further clarification, please let me know.