To attempt or not to attempt - A risk management approach toward negative marking

1.0 Introduction

In some of the competitive examinations, there is negative marking. When equal weightage is given for both right answer and wrong answer, the risk of losing marks when a wrong answer is given is high. In case of multiple choice questions, there will be always some probability of answering the question correctly, which increases when there is more than one right answer.

In risk analysis, we have the following formula for the expected value of a variable X.

E[X]=Sp_iX_i  for all possible values of X=X_i,

where 

E[X]=expected value of X.

p_i = probability of the value of X being X_i

2.0 Joint Entrance Exam Main (JEE Main) - A case study

2.1 Introduction - JEE Main

In India, we have the famous Joint Entrance Examination (JEE) for admission into various engineering, architecture and planning courses across the nation offered by various academic institutions run by or funded by the Government of India. This is conducted in two phases i.e. JEE Main and JEE Advanced. JEE Main is the basic qualifying examination for the said courses.

For 2020, the information bulletin for JEE Main can be found at the following link:

https://jeemain.nta.nic.in/WebInfo/Handler/FileHandler.ashx?i=File&ii=58&iii=Y

From the above bulletin (see pages 6 and 7), we find that every correct answer is awarded 4 marks and every wrong answer is awarded one negative mark (-1).

2.2 Definition: Uncertain question

Let us define an uncertain question as the MCQ for which the examinee doesn't know the right answer.

2.3 Answering uncertain questions at random

Going by the question paper for 2019, there are four options for each multiple choice question (MCQ).

The probability of choosing the right answer at random is 1/4=0.25.

The probability of choosing the wrong answer at random is 3/4=0.75.

So, the expected value of attempting a question by randomly choosing an answer is 0.25x4-0.75*.1=0.25.

This means, one may still score positively by taking a chance. However, the probability of the answer being wrong is very high.

When four questions are wrongly answered and one question is rightly answered out of five questions, the outcome is 4*(-1)+4=0. From this, it is evident that the risk of losing marks happens only when one scores less than 1/5th of the total number of questions where one is uncertain of the answer.

When p is the probability of answering a question correctly, 1-p is the probability of answering it wrong. In this case, p and 1-p are 0.25 and 0.75 respectively.

The probability of answering r answers or less correctly for n questions is,

When n= 1 to 5, negative marks will be scored only when i=0.
The probability of obtaining negative marks for n=1 to 5 is as follows:

n	P
5	0.237
4	0.316
3	0.422
2	0.563
1	0.750

Given that there are 20 MCQs in each question paper, the probability of obtaining negative marks by answering all questions at random occurs when less than 4 answers are right i.e. when the number of right answers is 3 or less.

The probability of obtaining negative marks when 20 questions are answered at random is P=0.225.

From this, though the probability of scoring positive marks or zero out of all uncertain questions is higher except when the no. of MCQS are one or two, there is still a good amount of risk involved. Whether one can take this much risk or not, depends on the future plans one has.

2.4 Making intelligent guesses for MCQs

The risk of scoring negative can be minimized by guesstimating the answers that appear to be the most right.

Assumption: Let us assume the probability of choosing the right answer increases to p=0.5 when one makes an intelligent guess combined with subject matter knowledge.

In this case, the estimated value of attempting one uncertain answer is 0.5*4-0.5*1=1, which is much higher than before.

Now, the probability of a negative score when 8 uncertain questions are answered is,

P=0.035 which is statistically insignificant, as statistical signifcance is often attached to a confidence level of 95% (probability of 1-0.05) and more. This means, we can say with a confidence level of 0.965(1-0.035) or 96.5% that will either improve one's score or there is no harm at least, by making intelligent guesses when the no. of uncertain questions is 8 or more. 

Even when there is only one uncertain question, the probability of scoring a negative mark is 0.5 only if the assumption made above is right. For a higher number of uncertain questions, it will be less than 0.5 which means the probability of net benefit or no loss out of the whole exercise is higher.

3.0 Applicability of the case study

The above case study is applicable in similar cases when the following conditions are met

1.  When the options provided are four out of which only one is correct. When more than one option is correct, however, the risk comes down.
2. The ratio of positive marking to negative marking is 4:1.

4.0 Conclusions

1. For the above case study, it is shown that the expected value from taking the risk of attempting uncertain questions is always positive.
2. The probability of scoring positive is higher than the probability of scoring negative, when more than two multiple choice questions are answered at random.
3. The probability of scoring positive is higher than the probability of scoring negative when more than one multiple choice question is answered by making an intelligent guess, when one can guesstimate correctly with a probability of 0.5.
4. The risk of scoring negative marks comes down when one makes an intelligent guess of the answer rather than selecting the answer at random.
5. When 8 or more questions are answered by an intelligent guess, there is a very good possibility of scoring positive or suffering no loss from the guess work, when the probability of guesstimating the right answer is 0.5.

5.0 Path forward

1. One can do similar math when the number of options and/or the marking scheme are different.
2. The confidence level of scoring no negative mark can be recalculated by varying the probability of success for an intelligent guess.
3. Coaching centers can find out the probability of their students guessing the answer right in each subject separately and advise the students individually. As they have math and stat experts, they can even come up with better models (Good if they are already doing it, but I have not heard of any).
4. Coaching centers also need to teach and demonstrate how to make better guesses. Those preparing themselves for competitive exams on their own need to develop their own strategies.
5. When one has multiple options for pursuing higher studies, or multiple career options, one can take bigger risks, as bigger rewards may come from bigger risks.