negative marking

*made some important changes in 5-6 guesses*

Ok so for this module in my school, they employ this marking scheme for the MCQ exam called 'negative marking'. What this basically means is that for every correct answer we will get 1 mark, every wrong answer we will get -0.25 marks, and every blank we will not get any marks (or get 0 marks)

So comes the following question:
Would it better to leave any unsure questions blank, or simply guess all of them?
There should be a mathematical answer based on probability, so I shall set out to find this solution.

(all based on a 4-answer-choices MCQ)

The Basic Assumption
There is a basic assumption which we must make before we proceed, that is, there is a 1 in 4 chance of getting 1 mark, or 1/4 chance of getting 1 mark.
Some may be quick to point out that there are five options here, that is, 1 correct answer and 3 correct answers, AND the option of leaving it blank. Shouldn't it be 1/5?

The answer to that is simple, since leaving it blank is a definite 0 marks,(i.e. when you leave it blank, you KNOW for sure it is 0 marks, it is not a 1/5 chance that you will get 0 marks) it should be automatically excluded from the probabilities.

So with this basic assumption in mind, let's proceed.

Analysis
It is noteworthy that all the examples will just be weighing if it's better to guess all or to leave them all blank. They will not consider a 'mixture' of guessing and leaving blank, since that is a separate matter altogether.

So with the 1/4 assumption, it follows that
- for every 4 guesses, you will get 1 correct.
Since the 4 guesses are for 4 random questions, this has to be correct; it is mathematically sound.

Let's start with a simple example, there are only 4 unsure questions and hence only 4 guesses you make in the paper.
4 guesses - 1 correct, 3 wrong
-> +(1x1) -(3x0.25) = +0.25
Thus, more beneficial to guess all.

How about 5 guesses? This would be slightly more complicated, since 1/4 of 5 is 1.25, and rounding up or down would result in an inaccurate answer.
To get a round number, we take a group of 4 people with 5 guesses each. Thus 'getting 0.25 correct' means out of a group of 4 people, 1/4 of them will get it right (means they have TWO correct) while the other 3/4 of them will get only one right.
[Read the explanation in these square brackets if you don't get why it's 3/4 will only get 1 right.
Let's simplify matters to just 4 people having 1 guess each. If they each have a 0.25 chance of getting it right, it means 1 out of 4 will get it right.
So why that 1/4 group in the example above gets TWO correct is because their odds state they will get the 1st '1' correct, just that the 2nd '0.25' is up for probability]

Thus for 5 guesses, there's a 1/4 chance we will get 2 correct (better to just guess all)
and there's a 3/4 chance we will only get 1 correct (better to just leave all blank, as 1 correct in 5 means +1 -1.25 = -0.25)

Therefore, for 5 guesses, there is a higher probability that you will get a negative mark than a positive mark for guessing.

HOWEVER. What you should do here is to leave 1 of them blank (to guarantee 0 marks, no negative), and guess the other 4 (which will be the same case as 4 guesses, which is beneficial as shown earlier.)

6 guesses will follow the same logic, leave 2 blank, and guess 4.

7 guesses onwards it is more beneficial to go for the +0.75 (you have a higher probability anyway) than to settle for the +0.25 of 4 guesses, 3 blank. Thus:

7 guesses (instead of 7 blanks) will result in a higher chance of getting +0.75 than the negative case (work it out yourself), same for 8 and 9 guesses.
10 guesses is another 50/50 chance as in the 6 guesses case, but this time it's either a positive or neutral (0 net marks) result, so it's better to take your chance and guess them all too.

And by further deduction you can see that subsequent greater number of guesses will always result in a higher probability of getting a positive mark. Thus for 7 guesses and above, it is more beneficial to guess all than to leave all blank.


Now let's take a step back and examine the last 3 numbers which we did not look at - 1, 2 and 3 guesses.

1 guess
You have a 1/4 chance of getting 1 mark, and a 3/4 chance of losing 0.25 marks. It's a no-brainer, better to just leave it blank and guarantee no negation since the negative result has a higher probability.

2 guesses
We have to look at this in the case of multiple participants again, since 0.25 x 2 = 0.5
So out of 2 participants, it follows that either one will get it correct. It's another 50/50 case.
50% -> You get 1 mark
50% -> You lose 0.25 marks
This is the only ambiguous case, where the advantage of securing '0 points' by not guessing at all is not clear.

3 guesses
0.25 x 3 = 0.75
Which follows that there is a higher probability that you will be in the group that gets 1 /3 correct (3/4 chance to be exact) which is a +1 - 0.5 = +0.5 benefit, hence it is better to guess all for 3 guesses.


Summary
If you have 1 unsure question, it is more beneficial (or mathematically sound) to leave it blank.
If you have 3,4 unsure, guess them all.
If you have 5 or 6 unsure, leave blank 1 or 2 respectively, and guess the remaining 4.
If you have 7 or more unsure questions, it is more beneficial (or mathematically sound) to guess them all.
If you have 2 unsure.. flip a coin to decide whether you should guess or not, as there is no clear mathematical advantage on either side.


P.S.
Weiqin did a binomial calculation that more or less supports my noob calculations
http://i136.photobucket.com/albums/q166/atqhteo/31082010336.jpg
http://i136.photobucket.com/albums/q166/atqhteo/31082010337.jpg

ENJOY :D