Guessing on the SATs, is it ever better to leave it blank than to guess?

Question

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ points. If you answer no options, you receive $0$ points. You may not answer multiple questions.

It is said by the SAT test creators that leaving an answer blank is better than blind guessing, and I blindly believed them. I then thought, are they actually correct!

You see, the way I see, there is one answer for which you receive $1$ point and 4 answers for which you recieve $-\frac14$ points. Therefore in the long run, you would get $0$ on blind guesses, which is the same as the $0$ you get for not answering. Therefore guessing (blind or otherwise) is at least as good as leaving it blank in the long run no matter how much you know about the question.

Is this reasoning correct? I can't see a flaw in my reasoning, but its my argument against the authority of the SAT people. It seems more likely that my reasoning is a flaw, but I can not find it.

Note: Although this is not strictly necessary, but if my reasoning is correct, it would be good to cite reputable sources to build up a case to convince my school so they could inform students (people outside mathematics listen to authority, not reason.)

Answer 1

If points are awarded as you say, then indeed the expected points you can gain from choosing one of the five answers randomly is $$Ep=1/5*1-4/5*1/4=4/20-4/20=0.$$ However, you are missing the fact that this happens only in expectation, so it is not the case that

guessing (blind or otherwise) is always at least as good as leaving it blank.

You can put it another way: in 4 out of 5 cases guessing is worse than leaving it blank. Though fair enough, in 1 out of 5 you are better off.

Another point: people are usually risk averse, that is, would rather have 0 for sure than 0 in expectation, because the latter implies you sometimes get a negative score. But this is a matter of your preferences: if you are generally a gambler, then you might as well guess and gamble that you got the right answer.

Finally, as noted in the comments, as soon as you can reasonably rule out some options or can at least order the answers with respect to likelihood of being correct, then you should guess (but not blindly, i.e., pick the answer you think is most likely).

Answer 2

Always take the educated guess. You often can exclude one or two silly answers, and with chance 1 out of 3 you beat leaving it blank with a few points.

For example: what is the main chemical element on the moon?

1. Water
2. Cheese
3. Oxygen
4. Iron
5. Silicon

Well, if you know it's not cheese, and even Water is not an element, you can take a random pick out the 3 other options to beat the odds.

Answer 3

Your proof is right. In 4 out of 5 you are getting - 1/4 in 1 out of it is +1. This means mathematically speaking it does in the long term not make a difference if you guess or leave an answer blank. But that means it is exactly as good, not at least as good: on single test where you don't know any answers it might turn out worse or better.

I could think of some cases where it might following this thoughts be actually wise to guess: When your only chance to pass is luck. But you are usually not going to be in that situation.

Answer 4

You shouldn't guess for two reasons: 1-If you're wrong, you're giving proof that you haven't learned your lessons, rules, theorems... 2-You want to know your real level through SAT. If by guessing, you get some correct answers, then the score you'll have won't reflect your real value and skills.

Answer 5

Yes, if you have bad luck guessing then it would be better in that case to leave it blank. Since it is a SAT and not a SGT, they should penalize more for a wrong answer such as -1/2 so that if someone doesn't know an answer, they are less inclined to guess. Then they might get a better assessment of aptitude rather than guessing "skills" (actually luck). +1 for each correct answer, -1/2 for each wrong answer, and 0 for each blank answer would be my recommendation.