Your roommate Max is debating how much time he should spend with his tutor to get him fit for an upcoming exam. Can you help?
Max is facing an exam where he will have to nail 4 out of 6 tasks to pass. The tasks are drawn randomly from a pool of 100 different exercise topics. He can hire his knowledgeable tutor Soya, for $30 per hour. One hour is enough to cover 2 topics. Max did some studying throughout the semester. He is confident he can nail 50 topics. How many of the remaining topics should Max go over with Soya to minimize his costs if he can retake the course next semester for $1,000?
Looking at the expenses for this year and potentially next year, at what point does adding more topics mastered yield diminished returns?
The probability to pass exactly x tasks our of K (= 6 for us), when there is a total pool of N (= 100), when we know the answers to n, is given by the hypergeometric distribution.
Therefore, the chances that we nail at least 4 in our example is
Now, if we choose r as the number of additional topics we want to prepare for, then all we need to do is to minimize our costs. n = r + 50 in this case and we
Minimize 15*r + (1-P(X>=4)) * 1000.
Using Seeker results in r=26 for an expected pass probability of 85.24% and expected costs of $537.59 ($390 for tutoring and $147.59 for the expected costs of taking the course next year). Therefore, the recommendation is to book 13 hours of tutoring.
