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Multiple-Guess Tests
John Denker

## 1  Gaming the Test

### 1.1  General Suggestions

As mentioned in reference 1, if you are playing a game, know the rules of the game. If you are taking a test, know the rules of the test.

Multiple-guess tests are a rather silly sort of game. If at all possible, don’t put yourself in a situation where you will be judged by such a test ... but if you are stuck in such a situation, here are some hints on how to improve your score. These tricks only work in the artifical world of multiple-guess tests, and are in many ways the opposite of the real-world thinking skills that are the main topic of this document. However, sometimes you have to take tests, and if you’re smart you might as well seem smart by doing well on the test.

1. Choose the best answer, even if it is wrong, so long as it is less wrong than the others. That’s the rules.
2. For each question, read the stem of the question and the possible answers before starting to work the problem. That’s because:
• Often you can work backwards, starting from the possible answers.
• Sometimes you can eliminate N−1 of the possible answers on sight, without actually working the problem, or without working more than a tiny subset of the problem.

When following this process of elimination, be careful to check all of the answers. Don’t just eliminate N−1 of them and blindly accept the Nth one, because it is quite common to find that all of the answers are defective in some way. Remember: Choose the best answer, even if it is wrong.

3. Sometimes you can eliminate only one or two of the possible answers, in which case you should randomly guess among the remaining possibilities. This will improve your score on average. This is an obvious corollary of the rules. For more details on how the guessing strategy depends on the scoring rules, see section 1.2.

Some students have an emotional hang-up about this. They feel that guessing wrong is Wrong with a capital W, just like stealing is Wrong. It would be better to think of it like baseball: If you get a base hit one time out of three, it means you are a very skillful batter ... even though on a per-at-bat basis you fail twice as often as you succeed.

4. If there was any doubt or any guessing involved, fill in your best-guess answer (on the answer sheet) and then put a question mark next to the question (in the question book), so you can review it later.
5. Often, facts given in the stem of one question will give away the answer to another question.
6. If there is time pressure, go for the low-hanging fruit. On the first pass, answer the easy questions and skip the hard questions. If there is time left over at the end of the first pass, make a second pass. Re-examine the questions you marked as dubious, and/or work some of the hard questions.

### 1.2  Scoring Schemes; Risk and Reward for Guessing

As mentioned in section 1.1, very commonly there are situations where guessing will improve your score on average. This is an obvious corollary of the rules. Specifically, the scoring scheme must be considered part of the rules. Let’s look into this more closely.

Suppose there is a multiple-guess test with four possible answers for each question. Further suppose that your score on the test is equal to the number right minus 1/3rd of the number wrong.

• If you don’t answer any of the questions, the score is zero.
• If you write down completely random guesses, you will be right one quarter of the time and wrong three quarters of the time, so your score will be zero on average, since (1/4)*(+1) + (3/4)*(−1/3) = 0.

In this scenario, random guessing is a break-even proposition. On the other hand, if you can eliminate one of the possible answers, the 1-out-of-4 proposition becomes a 1-out-of-3 proposition, and it pays to guess, since (1/3)*(+1) + (2/3)*(−1/3) = 1/9 which is greater than zero. In this case the advantage is small, but if you do this enough times it adds up.

In general, for a multiple-guess test with N possible answers, the “normal” scoring scheme is the number right minus 1/(N−1) times the number wrong. In particular, for a true/false test, N=2 so the “normal” scoring scheme is the number right minus the number wrong, which seems harsh, but is necessary if you want completely random guessing to be a break-even proposition.

If you ever find a true/false test where there is no penalty for wrong answers, or if the penalty is less than a full point, it is to your advantage to guess at every question, even if you haven’t looked at the question. I’ve seen tests like this.

More generally, if you ever find a test with N possible answers where the penalty for wrong answers is milder than 1/(N−1), it pays to guess at every question. Do not leave any blank items on the answer sheet! I’ve seen tests like this.

## 2  References

1.
John Denker,
“Learning, Remembering, and Thinking”
www.av8n.com/physics/thinking.htm
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