Spider-Man Problem

[MilesProblem]

Hi, my friends and I were watching Spider-Man last night and there is a scene where Miles purposefully scores zero on a multiple choice test with 100 questions. The teacher points out the only way to get 0/100 is to know all of the answers.
We then got into a fairly heated debate about how the number of answer options affects the difficulty of scoring zero.
My argument, once I’d thought about it for a bit, was that the only way to guarantee a score of zero, is to know the correct answer. So scoring zero is the same as scoring 100%. Difficulty increases with the amount of options. So it is more difficult to GUARANTEE a score of zero, as the amount of options increases.
Can anyone who knows about maths confirm this is a better argument or tell me why I’m wrong?
Thanks

 

[Dr.Peterson]

The original claim is false. If you made random choices on a true/false test, you would have to be just as lucky to get 0% as to get 100%, but both are possible. And on a multiple-choice test, it is easier to get 0 than 100, because there are more wrong choices than right.

If you know all the answers, you can decide for yourself what score you want (if it is possible to get at all), and get it. But the only way to guarantee any particular score is to know all the answers.

On the other hand, if you don't, it is likely to be much easier to deliberately get 0 than 100, because each question may have a choice you can eliminate even if you don't know which one is right. Just pick one you know is wrong.

So your argument is backward.

 

[MilesProblem]

The original claim is false. If you made random choices on a true/false test, you would have to be just as lucky to get 0% as to get 100%, but both are possible. And on a multiple-choice test, it is easier to get 0 than 100, because there are more wrong choices than right.

If you know all the answers, you can decide for yourself what score you want (if it is possible to get at all), and get it. But the only way to guarantee any particular score is to know all the answers.

On the other hand, if you don't, it is likely to be much easier to deliberately get 0 than 100, because each question may have a choice you can eliminate even if you don't know which one is right. Just pick one you know is wrong.

So your argument is backward.

Thanks a lot for that. So the below is untrue?

- the only way to guarantee a certain score (0% in this case) is to know all the answers with certainty.
- the more choices there are, the more difficult it is to know with certainty which answer is correct
- so the more options there are, the more difficult it becomes to guarantee the score of 0%

We were trying to work on the premise that the person was going into the test with the guarantee of scoring 0%.

again, sorry if my explanation is convoluted. As you can see, I’m not of a mathematical mind!

 

[Dr.Peterson]

- the only way to guarantee a certain score (0% in this case) is to know all the answers with certainty.

Since "guarantee" means certainty, you can't guarantee anything you don't know; in this case, you can't be sure ahead of time that every question will contain a choice you know to be false. You have to know something with certainty!

What's definitely untrue is the claim that the only way to get a 0% score is to know all the answers.

- the more choices there are, the more difficult it is to know with certainty which answer is correct

But knowing with certainty is not necessarily a function of the number of choices. There are infinitely many possibilities for the answer to "2+2 = ?", but I am just as certain as I would be of the true/false question "2+2=4?" This claim is not about math so much as about the nature of knowledge. If you know, you know. If "distractor" answers distract you, you don't really know. But if you know something, but imperfectly, then more choices can make it harder to decide; and if you know nothing, then more choices make it less likely to get the right answer by chance.

- so the more options there are, the more difficult it becomes to guarantee the score of 0%

With more options, there is more chance that one of them is obviously wrong, so it becomes potentially easier to get 0%, if that is your goal.

But again, this is not really a probability question at this point.

We were trying to work on the premise that the person was going into the test with the guarantee of scoring 0%.

I'm not sure what this means. Perhaps you mean he's going in with the intention of scoring 0%, which is not the same thing unless you are also presuming perfect knowledge.

 

[Cubist]

But knowing with certainty is not necessarily a function of the number of choices. There are infinitely many possibilities for the answer to "2+2 = ?", but I am just as certain as I would be of the true/false question "2+2=4?" This claim is not about math so much as about the nature of knowledge. If you know, you know. If "distractor" answers distract you, you don't really know. But if you know something, but imperfectly, then more choices can make it harder to decide; and if you know nothing, then more choices make it less likely to get the right answer by chance.

If there's a question for which the answer is not directly known, then it's still possible to guarantee finding which is the correct multiple choice option by eliminating all except one of the options (eliminate all the options that are known to be false). I guess you could argue, subjectively, that this method of elimination would become harder if there are more multiple choice options.

For me, this seems the "weak link" in OP's argument. Because the more options there are, it becomes more likely that there will be at least ONE option present that is known to be false. Thus it becomes easier to guarantee choosing an incorrect answer if there are more options - you don't even need to know which option is correct in order to avoid choosing it, just pick the one that is known to be incorrect in order to guarantee 0% (if that is the goal)

 

[MilesProblem]

Since "guarantee" means certainty, you can't guarantee anything you don't know; in this case, you can't be sure ahead of time that every question will contain a choice you know to be false. You have to know something with certainty!

What's definitely untrue is the claim that the only way to get a 0% score is to know all the answers.


But knowing with certainty is not necessarily a function of the number of choices. There are infinitely many possibilities for the answer to "2+2 = ?", but I am just as certain as I would be of the true/false question "2+2=4?" This claim is not about math so much as about the nature of knowledge. If you know, you know. If "distractor" answers distract you, you don't really know. But if you know something, but imperfectly, then more choices can make it harder to decide; and if you know nothing, then more choices make it less likely to get the right answer by chance.


With more options, there is more chance that one of them is obviously wrong, so it becomes potentially easier to get 0%, if that is your goal.

But again, this is not really a probability question at this point.


I'm not sure what this means. Perhaps you mean he's going in with the intention of scoring 0%, which is not the same thing unless you are also presuming perfect knowledge.

thank you for your time!

 

[JeffM]

Thanks a lot for that. So the below is untrue?

- the only way to guarantee a certain score (0% in this case) is to know all the answers with certainty.
- the more choices there are, the more difficult it is to know with certainty which answer is correct
- so the more options there are, the more difficult it becomes to guarantee the score of 0%

We were trying to work on the premise that the person was going into the test with the guarantee of scoring 0%.

again, sorry if my explanation is convoluted. As you can see, I’m not of a mathematical mind!

Hi, my friends and I were watching Spider-Man last night and there is a scene where Miles purposefully scores zero on a multiple choice test with 100 questions. The teacher points out the only way to get 0/100 is to know all of the answers.
We then got into a fairly heated debate about how the number of answer options affects the difficulty of scoring zero.
My argument, once I’d thought about it for a bit, was that the only way to guarantee a score of zero, is to know the correct answer. So scoring zero is the same as scoring 100%. Difficulty increases with the amount of options. So it is more difficult to GUARANTEE a score of zero, as the amount of options increases.
Can anyone who knows about maths confirm this is a better argument or tell me why I’m wrong?
Thanks

So in your original question, the assertion was that the only way to get every question wrong is to know all the correct answers.

That assertion is FALSE. Suppose I ask you a question and you give the wrong answer. So according to your logic THE ONLY WAY TO GIVE A WRONG ANSWER IS TO KNOW THE RIGHT ANSWER. According to you, no one in the history of the universe ever gave a wrong answer on a test because they did not know the answer.

In fact, one way to get a wrong answer is because you do not know the correct answer and guess incorrectly. And if that applies to the first question, it applies to the second and then to the third and so on.

You then shift your ground and start talking about guarantees. John takes a quiz of three questions and gets all three questions wrong. Are we to conclude that it is guaranteed that John knew the correct answers to all three questions because he got them all wrong. The proposition is inane.

Mary has not studied for the quiz. She asks her friend Sue, who took the quiz in an earlier period and got all correct answers, what the answers are. Sue is mad at Mary because Sue had heard that Sue’s boyfriend had flirted with Mary. In a jealous snit, Sue writes down the wrong answers. But Mary trusts Sue and uses Sue’s wrong answers. Mary will get all wrong answers even though she does not know the correct answers. It is guaranteed because she trusted a woman scorned, for whom ****’s fury holds not a candle.

Now the true thought is that if you guess at random on enough questions the probability that you will not guess right on at least one of them is so low that for practical purposes we can say it is virtually impossible. The probability of guessing wrong on every true-false question out of just 20 questions is less than 1 in a million. So it is not truly impossible, but the practical way to go is to act as if it is impossible. The word you are looking for is “wildly implausible” rather than “impossible.”

The moral of this story is that you must not treat Spider Man as non-fiction.

 

[JeffM]

By the way, I failed to address the multiple choice dimension.

If you have a multiple choice (4 wrong 1 correct answer for each question) and 20 questions, the probability of guessing at random and getting 20 wrong answers is a little over 1%, definitely not impossible.With 50 questions, that probability drops to less than 1 in 10000, which gets us into the implausible category.