EV in my physics exam

[Josh11]

My physics exam has a quirk about partial credit. For problems with 5 answer choices, if you mark one answer and it is the correct answer, you get 6 points. If you mark two answers and one of them is the correct answer, you get 3 points. If you mark three answers and one of them is the correct answer, you get 2 points.

So if I'm not mistaken the EV of each choice works out the same. For example, if I have two answer choices and I pick one, I get six points once and 0 points once for a net of 3pts per question, which is the same as picking both answer choices. My question is, in which circumstances should I use the multiple answers and when should I only guess one? (ie narrowing down answer choices to two or three or just random guessing) Any thoughts?

Apologies if this is painfully simple, but being aware of the EV of the situation unlike most others makes me want to be sure what I'm doing is the best choice.

[Dromar]

Well, lets start with a simple example:

You're asked a question, and there are only two choices. If you choose one and are correct, you get 6 points. If you choose wrong, you get 0. If you choose both, you get 3 points for your correct choice, and 0 for your incorrect choice.

In the above simple example, you basically have two choices: try for the right answer, or just get 3 points automatically. The deciding factor is the probability that you estimate each of the given answers to be correct. If you give each of the two possible answers to the question 50/50 to be right, the EV is the same for both tactics.

6*(.5) = 3*(.5+.5) = 3

But, if you think that, for example, the first answer choice is anything greater than 50%, you get increased EV by choosing that answer over both (and obviously over choosing the one you think is wrong). If you were 70% sure the answer was A, then:

6*(.7) > 3(.7+.3)
6*(.7) = 4.2
you would choose A

So, thinking intuitively about the problem, it appears the way to get maximum EV is to:

Choose the LEAST amount of answers such that the sum of their probability of being correct is greater than the percent of options they make up.

So, you'll almost always choose just one answer: the one that's most likely to be correct.

What this means is, if you have five options A through E, and you estimate their probability of being correct to be 21%, 20%, 20%, 20%, 19%, respectively, you should choose A. This is obviously an extreme case, and the EV will barely be any different had you chosen A and B, or A, B and C, but it WILL be slightly higher.

Now, if the probabilities are 42%, 10%, 42%, 4%, 2%, you have a situation similar to the simple example above. You obviously won't choose B, D, or E. Whether you go with A, C, or both gives the same EV, just as found in the simple example.

In conclusion, since you would always choose one answer unless you decide they have equal probability, you should NEVER have greater EV choosing multiple answers, and only EQUAL EV when you decide that all probabilities are equal.

[Josh11]

Would there be merit in taking the sure thing by choosing both to avoid test "variance" of me getting unlucky and choosing more incorrect than correct?

EDIT: What I meant to say was, I think, would lowering my short term variance be a better choice so I don't get stuck with an unlucky bad result?

[ChromePony]

This is really just a personal preference, is it more important to you to be assured of a B or is it worth the risk of getting a C to have a shot at an A...you get the idea.

[fiskebent]

I assume that the result of the test will be a grade and not a point score?

If so, then you might take into account how the points are converted to grades and what your 'correct' score would be. By correct I mean the average point score you'd get over a number of tests.

If the points are converted to grades like this (assume max. points is 100):
91-100: A
71-90: B
51-70: C
31-50: D
11-30: E
0-10: F

If you think your average score would be 88, it'd make sense to take some risks to get a score of 91 or better. You'd have to be pretty unlucky to get a C and just a little lucky to get an A.

I'm sorry if the grade scale makes no sense. I live in Denmark and we have a numeric grading scale instead of the letters used in the US.

[WarDekar]

UofI?

[Josh11]

[ QUOTE ]
UofI?

[/ QUOTE ]

Yes.

Thanks for the replies all.

[Jouster777]

If you can stratify answers such that one is more likely than others then always choose just that answer to maximize EV (though variance will increase).

If you can exclude some but the others are all equally likely then EV won't change depending on your choices...as long as you never mark answers you know to be wrong your EV is the same. All that matters is how much you like variance...fewer marked = more variance.

[mindflayer]

Dormar is right,
The choices only have = EV IF you are a monkey that is randomly choosing an answer. (monkey = person who has no knowledge of the subject even to make a logical guess.)

If you did any study or reading you must be able to determind that One of the answers is more likely than one of the others to be the right answer, therefore the EV of one, two or three marked answers is NOT equal.

[AaronBrown]

You've had some excellent answers so far. I'd add a point to fiskebent's response.

If you pick two or more answers to every question, the best grade you can get is 50%, which I assume is failing. So you have to answer at least some of the questions with only one answer. In fact, to have any reasonable chance of a good grade, you probably have to answer almost all the questions with one answer. Also, if there are a lot of questions for which your best answer has less than a 50% chance of being correct, whether you pick one answer or many equally good ones is unlikely to make much difference. For example, if you answer 30 questions with a single answer that has 1/3 chance of being right or two answers that have 2/3 chance between them; you expect to get 60 points with a standard deviation of 15 points either way. It's only for a small number of questions that the distribution of points per question matters much.

So the only question is what do do with a few questions for which one or more answers seem about equally likely to be correct as your best guess. Once you get to that point on the exam, you probably know about how many questions you have correct, and can weigh whether to play it safe by spreading your answers around, or taking more of a chance by giving single answers.

An important question is how disputes are handled. Suppose the answer to question 1 is supposed to be A. Someone successfully argues that the question is ambiguous and B could be correct. Presumably, anyone who gave a single answer of A or B would get full 6 points credit, as would someone who gave two answers, A and B. But what of someone who answered (say) B and C? Would they get zero points (did not get the correct answer and have no argument because they clearly didn't see the ambiguity), one and a half (got half a half right answer), three points (half of the B six point score) or six points (got a correct answer and may have been confused by the ambiguity)? This could make a big difference to the expectation.