To Guess or Not to Guess?

Should you try to guess an answer to a multiple-choice problem during a test? How many problems should you try to guess? I will talk about the art of boosting your guessing accuracy in a later essay. Now I would like to discuss whether it makes sense to pick a random answer for a problem at AMC 10.

Let me remind you that each of the 25 problems on the AMC 10 test provides five choices. A correct answer brings you 6 points, a wrong answer 0 points and not answering at all gives you 1.5 points. So guessing makes the expected average per problem to be 1.2 points. That is, on average you lose 0.3 points when guessing. However, if you are lucky, guessing will gain you 4.5 points per problem, and if you are unlucky, it will lose you 1.5 points per problem.

So we see that on average guessing is unprofitable. But there are situations in which you have nothing to lose if you get a smaller score and a lot to gain if you get a better score. Usually the goal of a competitor at AMC 10 is to get to AIME. For that to happen, you need to get 120 points or be in the highest one percentile of all competitors. This rule complicates my calculations. So I decided to simplify it and say that your goal is to get 120 points and then see what mathematical results I can get out of that simplification.

First, suppose you are so accurate that you never make mistakes. If you have solved 20 problems, then your score without guessing is 127.5. If you start guessing and guess wrongly for all of the last five questions, you still have your desired 120 points. In this instance it doesn’t matter whether you guess or not.

Suppose on the other hand that you are still accurate, but less powerful. You have only solved 15 problems, so your score without guessing is 105. Now you must be strategic. Your only chance to get to your goal of 120 points is to guess. Suppose you randomly guess the answers for the 10 problems you didn’t solve. To make it to 120, you need to guess correctly at least five out of the ten remaining problems. The probability of doing so is 3%. Here is a table of your probability of making 120 points if you solve correctly n problems and guess the other problems.

n Probability
20 1.0
19 0.74
18 0.42
17 0.20
16 0.09
15 0.03
14 0.01
13 0.004
12 0.001

We can see that if you solved a small number of problems, then the probability of getting 120 points is minuscule; but as the number of problems you solved increases, so does the probability of getting 120 points by guessing.

The interesting part is that if you have solved 19 problems, you are guaranteed to get to AIME without guessing. On the other hand, if you start guessing and all your guesses are wrong, you will not pass the 120 mark. The probability of having all six problems wrong is a not insignificant 26%. In conclusion, if you are an accurate solver and want to have 120 points, it is beneficial to guess the remaining problems if you solved fewer than 19 problems. It doesn’t matter much if you solved fewer than 10 problems or more than 19. But you shouldn’t guess if you solved exactly 19.

If you are not 100% accurate things get more complicated and more interesting. To decide about guessing, it is crucial to have a good estimate of how many mistakes you usually make. Let’s say that you usually have two problems wrong per AMC test. Suppose you gave answers to 20 problems at AMC. What’s next? Let us estimate your score. Out of your 20 answers you are expected to get 18*6 points for them plus 5*1.5 points for the problems you didn’t answer. Your expected score is 115.5. You are almost there. You definitely should guess. But does it matter how many questions you are trying to guess?

The correct answer to one question increases your score by 4.5 with probability 0.2, and the wrong answer decreases your score by 1.5 with probability 0.8. One increase by 4.5 is enough for you goal of 120 points. So if you guess one question, with probability 0.2, you get 120 points. If you guess two questions, then your outcome is as follows: you increase your score by 9 points with probability 0.04; you increase your score by 3 points with probability 0.32; and you decrease your score by 3 points with probability 0.64. As you need at least a 4.5 increase in points, it is not enough to guess one question out of two. You actually need to guess correctly on both of them. The probability of this happening is 0.04. It is interesting, but you have a much greater chance to get to your goal if you guess just one question than if you guess two. Overall, here is the table of probabilities to get to 120 points where m is the number of questions you are guessing.

m Probability
0 0
1 0.20
2 0.04
3 0.10
4 0.18
5 0.26

Your best chances are to guess all the remaining questions.

By the end of the test you know how many questions you answered, but you don’t know how many errors you made. The table below tells you what you need in order to get 120 points. Here is how you read the table: The number of problems you solved is in the first column. If you are sure that the number of mistakes is not more than the number in the second column, you can relax as you made at least 120 points. The last column gives the score.

Answered problems Mistakes Score
19 0 123.0
20 1 121.5
21 2 120.0
22 2 124.5
23 3 123.0
24 4 121.5
25 5 120.0

If the number of mistakes you made is one more than in the corresponding row of the table, you should start guessing in order to try to get 120 points. Keep in mind that there is a risk: if you are not sure how many problems you solved already and start guessing, you might ruin your achievement of 120 points.

In the next table I show how many questions you can guess without the risk of going below 120 points. The word “all” means that it is safe to guess all the remaining questions.

Answered problems Mistakes Score Non-risky guesses
19 0 123.0 2
20 1 121.5 1
21 2 120.0 0
22 2 124.5 all
23 3 123.0 all
24 4 121.5 all

You can see that if your goal is to get 120 points, your dividing line is answering 22 questions. If you solved 22 questions or more, there is no risk in guessing. Namely, if you have already achieved more than 120 points, guessing will not take you below that. But if you made more errors than are in the table, then guessing might be beneficial. Hence, you should always guess in this case — you have nothing to lose.

Now I would like to show you my calculations for a situation in which you are close to 120 points and need to determine the optimum number of questions to guess. The first column is the number of answered questions. The second column is the number of mistakes. The third column is your expected score without guessing. The fourth column is the optimum number of questions you should guess. And the last column lists your chances to get 120 points if you guess the number of questions in the fourth column.

Answered problems Mistakes Score To guess Probability
17 0 114.0 8 0.20
18 0 118.5 7 0.42
18 1 112.5 7 0.15
19 1 117.0 2 0.36
19 2 111.0 6 0.10
20 2 115.5 5 0.26
21 3 114.0 4 0.18
22 3 118.5 3 0.49
22 4 112.5 3 0.10
23 4 117.0 2 0.36
23 5 111.0 2 0.04
24 5 115.5 1 0.20

You can see that almost always if you are behind your goal, you should try to guess all of the remaining questions, with one exception: if you answered 19 questions and one of them is wrong. In this case you should guess exactly two questions — not all that remain.

Keep in mind that all these calculations are very interesting, but don’t necessarily apply directly to AMC 10, because I simplified assumptions about your goals. It may not be directly applicable, but I hope I have expanded your perspective about how you can use math to help you understand how better to succeed at math tests and how to design your strategy.

I plan to teach you how to guess more profitably, and this skill will also advance your perspective.

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4 Comments

  1. Ravi B:

    It might be interesting to note that, if your sole goal is to get a 120 or better on the AMC 10, then it only makes sense to answer 21 or 25 questions. Any other strategy gets dominated by one of those two, not just is a probabilistic sense but in a universal sense. For example, if you have already answered 22 questions, then you might as well guess on the remaining 3 questions.

  2. Felipe Pait:

    The fact that you can improve your results by strategizing detracts from the value of the exam, because test-taking techniques is not something useful in “real life” that we would like students to learn. I understand that it is unavoidable in some types of competitions, but for the purpose of examining students I believe it is better to leave ample time.

  3. Anton Lebedev:

    When I took the AMC 12, I just answered the questions I was certain about. I didn’t care about scoring high; I just wanted to secure a spot on the AIME.

  4. Tanya Khovanova’s Math Blog » Blog Archive » How to Boost Your Guessing Accuracy During Tests:

    […] promised to discuss how to improve the accuracy of your guessing at AMC 10/12, or other tests for that matter. There […]