Teacher: You got a 0.
Miles Morales: Is that...failed?
Teacher: Listen. If you answer all questions on a multiple choice test randomly, you would get a 25%. This means the only way for you to answer all questions wrongly is to...know the answer to all of them. Good job.
- From Spider-man: Into the Spiderverse.
Today, you had a probability exam. The teacher, wanting to spice up things, offered you another choice:
If you answered $k \%$ of the questions correctly, and $k$ is less than or equal to a predetermined value $p$ ($p < \frac{1}{number\ of\ answers}$), then the professor would assume that you answer the questions wrong on purpose, and your exam score would instead be $100 - k$ (For example, if $p$ is $20\%$, and you answered $10\%$ of the exam correctly, then your exam score will be instead $100 - 10 = 90$ out of $100$. If you answered $21\%$ of the exam correctly however, then tough luck, because your score would just be $21$ out of $100$).
You, knowing this, looked at all the questions first. You noticed the following thing:
- There are $m$ multiple choice questions, each has $n$ different answers $(n \ge 2)$. Only one answer is the correct answer for each questions.
- There are $X_1$ questions that you know exactly what the correct answer is.
- There are $X_2$ questions that you can eliminate down to $2$ answers.
- There are $X_3$ questions that you can eliminate down to $3$ answers.
- $...$
- There are $X_{n}$ questions that you can eliminate down to $n$ answers i.e. to you, any answer could be the correct answer. $X_1 + X_2 + ... + X_n = m$
The question is, what combination of $p$, $m$, $n$ and $X_i$ is needed for you to pursue the strategy of answering questions wrong on purpose and get the best expected score?