# Deciding a strategy to maximize my score in this test.

I will be taking a multiple choice test in two months. There will be 140 questions (5 possible answers): each right answer gives me +1 point, and each one I get wrong resuts in -0.25 points.

At this time I'm convinced that guessing questions where I can exclude at least 1 possible option is the strategy that gives me the highest absolute expected score even though I realize it has a high variance. Is this right?

Consider that to get the job I need to finish let's say in 90-95th percentile and not just beat the mean.

Is it worth using a different strategy (not answering if I'm not 100% sure, guess just the questions I'm undecided about 2,3 options?).

At this time if I have to generalize my average performance out of 140 questions:

• 80 questions: I'm 90% sure I chose the correct answer.
• 20 questions: I have to guess between 2 answers (let's say I get it right 60% of the times).
• 20 questions: I have to guess between 3 answers (let's say I get it right 35-40% of the times).
• 20 questions: I have to guess between 4 or 5 answers.

I think I need to score between 110-120 points. And of course I hope to improve my performance as I keep studying.

Thanks for any input.

• Don't think there's enough information. The best strategy is to write down all the correct answers, though presumably that one is not available. What distribution do you have? If, say, you expect to know the correct answer $90\%$ of the time then best is to just answer those. If you never expect to know the right answer then you have to guess a lot. – lulu Oct 3 '17 at 20:09
• @lulu I've added some more details in the question. Thanks for your kindness. – KingBOB Oct 3 '17 at 20:25
• Well, with this distribution I'd say you have to guess a lot. Even the last group. If you tried to get by with just the first three groups your expected total is $80+\frac 12\times 20\times 1 -\frac 12\times 20 \times .25 +\frac 13\times 20 \times 1- \frac 23 \times 20 \times .25 = 90.83333333$ so you will need a lot of luck to get to $110$. – lulu Oct 3 '17 at 20:43

## 2 Answers

If you can eliminate one answer choice, you have a 1/4 chance of earning +1 point, and a 3/4 chance of losing -0.25 points. Thus, on average, for every four questions where you can only eliminate 1 answer choice, you have a net gain of +0.25 points. Obviously, your chances are even better the more answer choices you can eliminate. If you can narrow your choices down to two questions, you have an average net gain of +0.75 points every two questions.

If you cannot eliminate any answer choices, you can expect, at best, a net gain of 0 points every five questions. So it's best to skip these questions.

Also, studying will help in two ways: you'll likely encounter fewer questions where you have to guess, and you'll probably be able to eliminate more choices on average when you do have to guess.

You could think about this through expected value.

In the cases where you're guessing from 2 answers or from 3 answers, you might as well guess, because you're expected to gain points in expectation. Even though you say you get the questions where you have to guess from 2 answers right 60% of the time, let's compute the expectation with 50%, which represents a random coin flip to decide the answer. For a single question, then the expected value when you have to guess from 2 answers is $0.5 * 1 + 0.5 * -0.25 = 0.375$. Meaning for 20 questions, you're expected to gain $20 * 0.375 = 7.5$ points. Using similar logic, we can see that the expected value when guessing from 3 or 4 answers is also positive, meaning that with just random guessing you're expected to have a net gain. Meanwhile, when guessing from 5 answers, you're expected to not gain or lose points (expected value of 0).

I would say then that if you can cross out at least 1 option, it may be reasonable to guess.

One other thing worth checking out is your expected score from the probabilities you gave. Assuming you answer every question for the 80 that you're 90% sure you chose the correct answer, you're expected to get $80 * 0.9 * 1 + 80 * (1 - 0.9) * (-0.25) = 70$ points. For the 20 questions you have to guess between 2 and there's a 60% chance you get it right, you're expected to get $20 * 0.6 * 1 + 20 * (1 - 0.6) * (-0.25) = 10$ points. Using similar logic with questions you have to guess between 3 answers, and a 40% success rate, you're expected to get 5 points. This totals only 85 points so far, and you have 20 questions left, which you're all picking from 4 or 5 answers for. With your current average performance, it's hard to see you scoring above 110, so I think more preparation needs to be done.

Overall, the optimal strategy does seem to be studying more, and with regards to the actual test, guessing if you can eliminate at least 1 option. Do note that this isn't a steadfast rule you have to go by and rather just a guideline. If you come across a question where you can't eliminate any choices but are very confident that one of them is more likely to be the answer, it might make sense to guess, so consider the guideline but ultimately use your best judgement. Good luck with your exam!