## Mathematics behind guessing answers in MCQ

Many times I've seen people saying that because of negative marking, one should only attempt questions in objective examinations that one is fully sure of.
I've seen this advice being given for the IIT-JEE, the AIEEE, and even the UPSC exams (negative marking has been introduced ).
This advice is WRONG, WRONG, WRONG.
Why am I saying so? It's all a question of probabilities. Read on to find out.
Why are negative marks given?
Before negative marking was introduced, several students got through just by guessing questions blindly.
Negative marking is meant to discourage blind guessing.
The keyword is BLIND.
Negative marking is NOT meant to discourage intelligent guessing.
Let's take the example of the IIT-JEE/AIEEE straight (one option correct out of four) questions (+3/-1). Say, there are 60 questions. You do not know the answers to any of them.
1. Guess blindly
Since one out of four options is correct, odds are, you'll get one out of every four questions correct.
That means that you should get around 15 questions correct, and 45 incorrect. So your total will be around (15 × 3 - 45 × 1) = 0, rendering you almost equal to a person who left the paper blank.
This is how negative marking discourages blind guessing.
2. Eliminate 1 option in every question
Let's say you know enough to eliminate one option in every question. Therefore, you have to guess among 3 answers. So the odds are that you get 1 out of every 3 answers correct.
This means that you should get around 20 answers correct, and 40 wrong. The total will be around (20 × 3 - 40 × 1) = 20. This means that you will be better off than if you had left the paper blank.
3. Eliminate 2 options in every question
Your odds will improve further. You should get around 30 answers correct, and 30 wrong. The total will be around (30 × 3 - 30 × 1) = 60. Your score is MUCH better than if you had left the paper blank.
Moral of the story
For a +3/-1 question, ALWAYS guess once you have eliminated at least one answer. ALWAYS.
Note that this is valid for all multiple choice questions in which the negative mark for a wrong answer is 1/(n-1) times the mark for a right answer. n is the number of options.
The situation can change according to the value of the fraction.
For the +4/-1 comprehension questions (in this year's IIT-JEE - 2009/2008/2007):
Even if you guess blindly, in 60 questions your score will be around (15 × 4 - 45 × 1) = 15. Therefore you should ALWAYS attempt such questions, even if you have to guess blindly.
For the +5/-1 multiple answer questions (previous IIT-JEEs):
There are 14 possible incorrect answers and only one possible correct answer. In 60 questions, if you guess blindly, you will get (4 × 5 - 56 × 1) = -36, i.e. badly hit.
If you are sure of one option being correct, there can be 8 possible answers -- you will get 1 out of 8 answers right. The odds are still against you.
If you are sure of 1 correct option and one other option (correct or incorrect), there can be 4 possible answers. The odds are *now* in your favour.
If you are sure of one option being incorrect, there can be 7 possible answers -- the odds are against you.
If you know two options are incorrect, there can be 3 possible answers -- the odds are in your favour.
Therefore, guessing is a bad idea, unless you are sure of two options either way.
Remember -- this is not a substitute for knowing the answers. You will always score the highest if you know the answers.
PS. It is possible that by chance you get all the answers you guessed right, or all the answers you guessed wrong. This chance is very small, however. To get all 60 questions wrong, the chances are (3/4)60 = 3.19 × 10-8. The probability of getting all 60 questions correct is even lower: (1/4)60 = 7.52 × 10-37.
EDIT: Some more information on this can be found at http://en.wikipedia.org/wiki/Expected_value . The concept of expected values is also present in gambling, and guessing is quite similar to gambling, in a mathematical sense. The important thing is to get the odds in your favour.

Man you speak the truth! I am in DCE ECE (secured 449 Rank) and I barely knew enough to cross cutoff. Then one of my friend from VMC told me to answer the questions you don't know with eene-meene or akkad bakkad stuff.... lol...
By the way that guy got rank under 30 in DCE CEE and this year he joined IITKgp CSE...lol

By the way you are good at probability!!... lol

wat u r saying is absolutely rite...
urs truly is a live example of this thing...
in an objective ppr..its nt necessary to knw th rite answr..
its imp to not hit the wrong answr..
no matter wats ur way of getting upto it..

Well I'm in IIT Bombay, and I certainly agree with you. Not that I guessed a lot of questions in jee but it certainly improved my rank marginally. Anyways nice blog!!

@Dhawal

Nopes dude, I did not go about guessing questions blindly, but now I wish I did. Few days ago I did an experiment and marked 160 questions randomly and managed to score 128 lol

he he he, it works! I know. A lot f people do it. Btw you seem to be good at math.

hi

i think u r wrong, i'm preparing for jto exam. it has 120 questions with 4 options.

due to ur calculation

if any one guess all questions correctly means

120 x 3 - 0 x 1 = 360.

but the exam was maximum 120 marks and minimum 35.5
negative marks: 1/3

plz explain soon coz day after 2morow i have exam.

mail me rohanr0055@gmail.com