A student, Kevin, has handed in all her coursework for CS2114, and wants to workout the chance she will get a distinction in CS2114. She asked 300 students whohave had distinctions in the past, and 100 of them said they handed in all theircoursework. She then asked 600 students who didn’t get distinctions in the past,and only 50 of them said they handed in all their coursework. Past experience showsthat only 15% of students who take CS2114 get distinctions(a) Calculate the prior probability of someone handing in all their coursework.(b) Using this prior and the information above, calculate using Bayes’ rule whatthe probability is that she will get a distinction?(c) If she didn’t hand in all her coursework what is the probability of getting adistinction?
Question
A student, Kevin, has handed in all her coursework for CS2114, and wants to workout the chance she will get a distinction in CS2114. She asked 300 students whohave had distinctions in the past, and 100 of them said they handed in all theircoursework. She then asked 600 students who didn’t get distinctions in the past,and only 50 of them said they handed in all their coursework. Past experience showsthat only 15% of students who take CS2114 get distinctions(a) Calculate the prior probability of someone handing in all their coursework.(b) Using this prior and the information above, calculate using Bayes’ rule whatthe probability is that she will get a distinction?(c) If she didn’t hand in all her coursework what is the probability of getting adistinction?
Solution
(a) To calculate the prior probability of someone handing in all their coursework, we need to find the proportion of students who handed in all their coursework among all the students surveyed.
The total number of students surveyed is 300 + 600 = 900.
Out of the 900 students, 100 of them who had distinctions in the past said they handed in all their coursework. Therefore, the proportion of students who handed in all their coursework is 100/900 = 1/9.
So, the prior probability of someone handing in all their coursework is 1/9.
(b) To calculate the probability of Kevin getting a distinction using Bayes' rule, we need to consider both the prior probability and the likelihood.
The prior probability of Kevin handing in all her coursework is 1/9.
The likelihood of Kevin getting a distinction given that she handed in all her coursework is 15% or 0.15.
Using Bayes' rule, the probability of Kevin getting a distinction is calculated as follows:
P(Distinction | Handed in all coursework) = (P(Handed in all coursework | Distinction) * P(Distinction)) / P(Handed in all coursework)
P(Handed in all coursework | Distinction) = 100/300 = 1/3 P(Distinction) = 0.15 P(Handed in all coursework) = 1/9
P(Distinction | Handed in all coursework) = (1/3 * 0.15) / (1/9) = 0.45
Therefore, the probability that Kevin will get a distinction, given that she handed in all her coursework, is 0.45 or 45%.
(c) If Kevin didn't hand in all her coursework, we need to calculate the probability of getting a distinction under this condition.
The probability of not handing in all coursework is 1 - 1/9 = 8/9.
Using Bayes' rule again, we can calculate the probability of getting a distinction given that Kevin didn't hand in all her coursework:
P(Distinction | Not handed in all coursework) = (P(Not handed in all coursework | Distinction) * P(Distinction)) / P(Not handed in all coursework)
P(Not handed in all coursework | Distinction) = 200/300 = 2/3 P(Distinction) = 0.15 P(Not handed in all coursework) = 8/9
P(Distinction | Not handed in all coursework) = (2/3 * 0.15) / (8/9) = 0.0375 / 0.8889 ≈ 0.0423
Therefore, the probability of getting a distinction, given that Kevin didn't hand in all her coursework, is approximately 0.0423 or 4.23%.
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