Multiple Choice Question negative marking

[I_am_learning]

You attend a multiple choice question exam, and you have n (say 10) questions whose answers you don't know at all.
There are 4 choices in each question.
A correct answer yeilds 1 marks.
An incorrect answer has penalty of -0.25 marks.
Is it wise to attempt all the questions in random? It seems wise to me.
What if n = 3 (or 1).
There are thousands of student participating (no-one knows the answers . You have to stand above maximum of them.

 

[Stephen Tashi]

You have to stand above maximum of them.

Is that requirement part of the statement of the problem? If this is a problem that you are inventing then there are many interesting ways to fill in the details. There has to be a precise definition of what "the answer" is before particular mathematics can be done.

For example if the goal is maximize the probability that you score is above the max score of 2000 other students then I agree with your intuition that you have a better chance (although a small one) by guessing at some questions rather than leaving all the questions unanswered.

I assume n is the number of choices per question.

 

[I_am_learning]

Hi,
I am trying to invent the problem. So, by the 'You have to stand above ...", I simply meant, you want to do your best. :)

n is the total no. of questions available. No. of options is always 4.

My original thought was,
The expectation of Marks to be earned by solving each question is 0.0625. (-0.25 * .75 + 1*.25), which being greater than 0 seems to be advantage.
So, if you have a bunch of such questions, you are likely to earn some marks by attempting them.

But, what if n = 1? (only one question)
Although, the expectation of Marks earned E(M) = 0.0625, still greater than 0,
it would be a foolish job to make the guess at random because the odds of loosing is 3/4 against 1/4 of winning.

So, what is the transitional value of n, above which guessing is favorable?

Also, what if there are some 'm' questions which carry 16 marks for correct, -4 marks for wrong answers, mixed-up in the original 10 questions?

 

[Stephen Tashi]

You still need to define what you mean by "do your best". It appears you mean that you want the strategy with the highest expected score.

 

[I_am_learning]

Yes, you want as much score as you can. Take that as a requirement. :)

 

[Stephen Tashi]

Yes, you want as much score as you can. Take that as a requirement. :)

"Yes" is a clear answer. However maximizing the expected score may not be the same as getting "as much score as you can", depending on how we interpret that statement.

But, what if n = 1? (only one question)
Although, the expectation of Marks earned E(M) = 0.0625, still greater than 0,

So n is the number of questions, not the number of choices per question?

 

[Zula110100100]

you have n (say 10) questions

From the OP

n is the total no. of questions available. No. of options is always 4.

From #3

 

[Zula110100100]

I believe you still want to guess since when n=1, E(n) = .0625, which directly means you have a 3/4 chance of missing, and losing .25points, but a 1/4 chance of gaining 1 point, so while it is more likely you will not get it, the points gain from getting it make up for that? Still comes out to a possibility of .0625 points versus 0 points.

 

[I_am_learning]

Zula, Think in practical scenario. (I am talking n=1)
You have only one Question.
Only one chance to answer.
You are very likely to too loose, (3/4), why would you make a guess?
Although the reward is great it has slim chance.
My mind is sort of twisted now. :)

 

[Stephen Tashi]

The feeling you have illustrates why you must define your objective precisely. You feel that there is a high probability of losing and you are now setting your goal as "not to loose" or "not to have a high probability of losing". But this is different than the goal of "maximizing my expected score". Since you are inventing the problem, it's your choice what goal to set, but when you set different goals, you may get different answers.

 

[chiro]

This kind of problem seems like a good use for Bayesian statistics.

Aside from that there are a quite a few ways to tackle this.

In one instance if the material does not cover all of the material in a uniform way, then chances are if you nailed that area, your score would be high, and if you didn't go so well, you will probably lose marks. Your subjective prpbabilities could incorporate this.

Another thing has to do with probabilities of how many right answers are A's, B's and so on. If we expect uniform distribution in this regard, this will affect the probabilities involved in a different way.

There is probably a myriad of other possibilities to consider, but it is important to think of things like this to help get a more accurate probability assessment.

 

[I_am_learning]

The feeling you have illustrates why you must define your objective precisely. You feel that there is a high probability of losing and you are now setting your goal as "not to loose" or "not to have a high probability of losing". But this is different than the goal of "maximizing my expected score". Since you are inventing the problem, it's your choice what goal to set, but when you set different goals, you may get different answers.

Why are they different things? Sorry I couldn't understand.

Anyway, which objective do you think is more suitable? Suppose that it is a job Competetive Job interview question, and you want the job. :) You can Fill in the details if its missing.

 

[Stephen Tashi]

Why are they different things? Sorry I couldn't understand.

It might be better to ask why those goals should all be the same thing! The numerical example that you have in front of you (for n = 1) demonstrates that the goal of maximizing expectation implies a different course of action than the goal of avoiding a high probability of losing.

An intuitive way to understand this is that "expectation" is not necessarily "what we expect to happen". It is property of a probability distribution and you can't, in general, "expect" the expected value to be what happens when you take 1 sample from that distribution.

Anyway, which objective do you think is more suitable? Suppose that it is a job Competetive Job interview question, and you want the job. :) You can Fill in the details if its missing.

I don't see how to phrase the problem realistically as a job interview. Do we assume the applicant can decline to answer? Do we assume he can evade the question and get a neutral score? Is there only one job being offered? Do we assume there are other applicants who have answered (if only by guessing) the question correctly?

A significant part of inventing this problem is how to treat a low score. For example, if we pretend the test is some sort of standardized test that many evaluators will see then getting a low score could have a lasting bad effect on your career. If it is a test given by only one employer and the results are known only to that employer, then you could risk getting a low score without lasting consequences.

Another significant part of inventing the problem is whether the "utility" of the score varies with its size or whether the utility of a score depends on its relative size among another set of scores. For example, you could treat answering question as a gambling situation where you loose $25 for a wrong answer and get $100 for a right answer. Then it doesn't matter to you how other gamblers do on "the test".