# How can I maximize my chances on an exam?

I have an exam tomorrow, there are 18 topics, the professor gives only one. But it is not enitrely random how he gives the topic.

First he gives an interval like 18-52, 30-62, 30-68 etc, then we both independently write down a whole number from this interval, and the sum of these numbers modulo 18 will be the topic one has to talk about.

Can I somehow affect my chances, like I say the middle of the interval, or the end of the interval?

• Which of the 18 topics do you feel you know best? If you are equally prepared to be examined on all of them, it does not matter what number you choose, does it? – Dilip Sarwate Jun 1 '15 at 13:35
• When I was a student we had the cards laying on the desk. You picked any card and it had questions and problems to work on. It was random. The strategy is simple: assume that any question may be drawn, and prepare for all of them. – Aksakal Jun 1 '15 at 13:57
• @DilipSarwate I'm not equally prepared, I know the last ones the least – user137425 Jun 1 '15 at 14:02
• @whuber We both independently write the numbers down. – user137425 Jun 1 '15 at 16:44
• @whuber I think they are not predefined, the ones I wrote in my question were the intervals last week. – user137425 Jun 1 '15 at 16:58

Since you do not know the number that will be written by the professor, you have to assume that he can write any of them. The following approach offers a way how to know all possible questions, given your action.

Let your number be denoted $a$ while the professor's $\omega$. The set of questions you can be asked, for given $a$ is $$Q_a =\{i=(\omega+a)\%18|\omega\in I\}$$ where $I$ is the considered interval.

More information could be obtained if you would know the strategy how the professor writes $\omega$. Assuming that it is a random number with probability mass function $p(\omega)$, you can transform the $Q_a$ to conditional probability mass functions $p(i|a)$: $$p(i|a)=\sum_{\omega\in I:(a+\omega)\%18 =i}p(\omega)$$ Afterwards, you can find $a$ that maximizes your chances $$a^{*}=\arg\min_{a} p(i|a) z(i)$$ where $z(i)=0$ if you do not know the topic and $z(i)=1$ if you know it.

If you do not know the professor's strategy, you can ask your classmates for his choices and to reconstruct the strategy on your own. Alternatively, you can try some common sense strategies (uniform, for example).

If the interval is randomly generated by your professor on the D-Day, I don't think there is a bias towards a specific topic.

The only chance you have would be to pre-define a strategy accepted by all your classmates. A general strategy may be hard to formalize if the interval is more narrow than 18. But with a large interval (like 19-37 for example), N-1 students has to pick a multiple of 18 (36 for this example) and the last one would pick a K such as K=18*k + t with t being the topic you collectively decided to target (K=20 in this example if you target the topic 2).

• This is not correct for the example intervals given in the question. For instance, the interval 30-68 contains the values 12, 13, and 14 (mod 18) three times each and the other values (mod 18) only twice each. If (for instance) the student prefers topics, 12, 13, or 14, they need only write down either 36 or 54, which are 0 mod 18. A random choice by the professor will then give each of 12, 13, 14 a 3/39 chance of occurring and all other topics will have only a 2/39 chance each. – whuber Jun 1 '15 at 16:54
• I thought OP just gave random examples. That's why I wrote "If the intervals are chosen randomly by your professor on the D-Day" at the beginning of my answer to avoid this misunderstanding. Nevertheless, I agree with your answer for these specific intervals. I edit now with "generated" instead of "chosen" for more clarity. – brumar Jun 1 '15 at 17:46
• It would depend specifically on what random distribution the professor chose for the intervals. If, for instance, she chose only intervals containing 18 consecutive values and (furthermore) chose uniformly at random within such intervals, then there is nothing one could do. But, conditional on the selected interval, there are useful strategies the student can follow to improve their chances of selecting a favored topic, provided only that the number of values in the interval is not a multiple of 18. This possibility appears to contradict both parts of your answer. – whuber Jun 1 '15 at 17:58
• @whuber I understand and agree with your point. That's why my answer is explicitly framed under the case of total uncertainty on this generated interval the day of exam, which seems to be the case according to the details you asked to OP. – brumar Jun 1 '15 at 18:20
• I appreciate that, but I still cannot see how your answer is correct. Even when one is completely uncertain beforehand about what the interval might be, once the interval is revealed there are strategies that improve upon your answer. I also cannot see how the strategies used by one's classmates have any bearing on the question: indeed, it makes no reference to classmates at all. I suspect you are reading more into the context than you have explicitly disclosed, which may be why your answer appears so problematic. Consider clarifying your interpretation as a preface to the answer. – whuber Jun 1 '15 at 18:35