Mastering MCQ Examinations

MCQ exams – they’re everywhere nowadays! More and more examinations are gradually going the MCQ way, and chances are that they’re going to play a role in your educational and professional life. You’ve probably either had to make your way through some of these exams, or will have to, or both. Either way, MCQ exams are playing an increasingly important role in courses, admissions, placements and recruitments.

Now, there are several factors that make MCQ exams quite different from traditional written examinations. Obviously, some of the standard principles still apply. You’ve still got to study hard and prepare as best you can. But if you understand and utilise the different practicalities and vagaries of these exams effectively, you will get a significant edge over the competition. From personal experience, I can tell you that the difference in marks can often be well over forty to fifty percent of what you would have otherwise achieved.

Now I’m going to explain, in as much details as it is possible to explain in such an article, what makes this possible for MCQ exams!

Firstly, let’s try to understand the basic distinction between written and MCQ exams. This is the distinction that makes everything possible.

Consider the following question :

When is my birthday?

Now suppose this is a written exam, and you don’t know the answer (which you won’t, unless you personally know me). Then you can’t answer the question, simple as that. If you took a random guess, the probability of you getting it right would be 1 in 365, assuming you don’t have to specify the year and you ignore February 29th(No, it’s not February 29th). If I remove those two restrictions, the probability becomes 1 in several thousand. So it’s basically, you know the answer or you don’t!

Now suppose this is an MCQ exam! Then you would be given the following options :

A) 21st October

B) 5th March

C) 14th December

D) 8th August

Now you suddenly have a 1 in 4 shot of getting it right. Pretty decent odds, right? Especially for something you probably have absolutely no idea about. You can just take a random guess, and there’s a pretty realistic chance that you’ll get it right.

So, what’s the distinction here? Well, if you haven’t figured out yet, it is this :

In MCQ exams, the question is given as well as the answer.

That’s right, you are being provided with the answer here as well. In a written exam, if you don’t know the answer to a question, there’s no way you’re getting it right. In an MCQ exam, that just isn’t true.

Now many of you would be thinking at this point, “Hey hold on! There’ll be negative markings more often than not, and you’ve got to be a bit lucky to get the correct answer.”

True, and true. Usually negative markings are included as a deterrent to indiscriminate guessing, and also there are a number of options from which generally all but 1 are incorrect. So this constitutes your risk. And if you do get lucky, the positive marks that you get – that constitutes your reward!

So, it ultimately comes down to a risk-reward equation. You take a risk, you stand to get some reward. You don’t take a risk, you don’t get any reward. But… are you leaving it all down to luck? You don’t want to do that after all your tedious preparation, right? Well, here’s what you should be doing:

Give yourself the best possible chance of getting the best possible result.

I cannot stress the above point enough. Yes, luck plays a factor here. Chance is involved. You need to be smart about the risks that you take, and the risks that you avoid. And if you manage to do that, chances are that your result would be significantly better than it would’ve been had you not taken any risks at all.

Unfortunately, there is no magic rule which will tell you exactly which risks to take and which to avoid. Each individual’s situation for each exam would be an unique case, and he/she needs to be the best judge of it. However, there can definitely be certain guiding principles to help you decide.

Now I’m dividing MCQ examinees into 4 broad classes. Of course, real life isn’t quite so deterministic and there’ll be outliers as well as variations within these classes themselves. However, they’re more than enough from an explanation standpoint. Keep in mind that in most MCQ exams, the goal isn’t simply to get the maximum possible marks out of 100(say), but rather to get a certain qualifying rank amongst the examinees, or sometimes to get more than a preset qualifying mark. Often, the case is such that, up to a certain rank, you could get into a certain institute, while up to a certain lower rank you could get into a slightly lesser institute and so on. So when we talk about results here, let’s think about these practical outcomes instead of just the actual marks (same logic applies even when the result is about marks).

I’ve obtained the following classification based on a candidate’s preparation levels, and expected results. Sometimes, a person’s expectations can change on seeing the question paper too. Ideally, that shouldn’t have much of an impact, but take that into account too. Remember that these classifications are theoretically based on just preparation levels, and assuming a 0 risk – 0 reward base level.

 

Class A – The candidate feels confident about obtaining his/her desired results based on his/her preparation levels. For example, the top 100 get into the best college, and he/she is confident of a double digit rank.

 

Class B – The candidate believes he/she will get some decent result. However, a better exam would result in getting a more desired option. For example, the top 10000 get into a variety of colleges, and the candidate feels he/she would definitely finish inside the top 5000 or so, but a better rank would result in a better college.

 

Class C – Based on preparation, the candidate isn’t very certain of getting the desired result. If the exam goes well, he/she might get it. Otherwise he/she might not. For example, the top 10000 get into a variety of colleges, and the candidate may or may not finish in the top 10000 based on the exam.

 

Class D – The candidate hasn’t at all prepared sufficiently for the exam, and is just appearing… just in case.

 

Clearly, as a candidate, you should take vastly different levels of risk, depending on which class you’re in. If you’re in Class A, you should take absolutely minimal risk, maybe even none at all. If you’re in Class D, you should be going for close to everything, at least give yourself some chance. If this was a standard written exam, a person in Class D would just stand no chance. However, this being an MCQ exam still gives a class D candidate some hope. Class B and Class C candidates should, generally speaking, be a lot more calculated with their risks. Also, in the case of class C candidates, the amount of risk to be taken could often vary sitting right in the exam hall. If the exam seems to be going better than expected, the candidate could cut down on the risks a bit, or if it is not going as well, the candidate could increase the risks he/she is taking.

One thing which should be becoming apparent now is that the lesser prepared students have even more to gain using such methods. The more risks you take, the more you stand to gain. And if you have less to lose, that gives you the license to take more risks. So the MCQ system could potentially act as a significant leveller. If you think about it, it’s logical that a system where it is possible to get the answer without actually knowing the answer is great for the times when you don’t actually know the answer. Of course, you always stand a better chance if you are better prepared, but here even with less preparation, you can still be in the game.

OK, so we’ve been discussing about exactly how much risk one should be taking. Now let’s shift the discussion a bit more towards evaluating the risks that you do take, figuring out the actual risk-reward ratio. Clearly this is a matter of calculating probability. But I’ll try not to get overtly mathematical about it. And you should too. You don’t want to end up spending half your examination time calculating probability figures that have nothing to do with your actual exam questions. Also, not everyone reading this would be equally adept at calculating probabilities.

While it is in no way essential or even beneficial to know the formal mathematics of probability to understand what I’m about to describe, an intuitive grasp of the concept of probability will definitely help in both the understanding and real time utilisation of these principles.

Here, we’re going to use the concept of an expected probability applied to an MCQ exam scenario. Here’s how it’s going to work – say you have 4 options, out of which 1 is correct. So, your probability of guessing the correct answer is 1 in 4. In other words, if you answer 4 such questions, it is expected that you will get 1 right and 3 wrong. Got that? Ok, now let’s introduce some marking scheme here – 3 marks for a right answer, and -1 for a wrong answer. So, answering 4 questions, it is expected that you get 3 marks for 1 right answer, and you get -3 for answering 3 incorrectly. This makes your expected probability 0 out of 4 questions i.e. you answer 4 such questions, you’re expected to get 0.

Now let’s suppose, out of the 4 options, you have somehow managed to eliminate 1. So you are now stuck between 3 options. So what does the expected probability become? Well, it’s now expected that you get 1 out of 3 correct. So the expected probability has become +1(+3 -2) out of 3 questions i.e. you answer 3 such questions, you’re expected to get 1.

Now suppose you’ve managed to eliminate 2 options, and you’re left with 2 options to choose from. So now it’s expected that you get 1 out of every 2 correct – an expected probability of +2 for every 2 questions.

Let’s analyse these results for a moment. The +3, -1, with 4 options is a pretty standard MCQ examination scheme, which also makes it a rather fruitful activity:

The first case where none of the 4 options are eliminated gives you an expected probability of 0 from 4(so basically, 0). This means that the risk-reward ratio is neutral. In other words, if you answer a number of such questions, you’d have to be rather unlucky to end up with a negative score. Similarly, you’d also have to be a bit lucky to end up with a positive score. Let’s call exams with such marking schemes neutral probability exams. A vast majority of MCQ exams are typically neutral probability. This is a very significant result as it indicates that in most typical MCQ exams, even if you randomly guess the answers of all the questions, you’re still not expected to end up with a negative score. Put another way, you’d have to be somewhat unlucky to actually lose marks by making completely random guesses on the exam. On the upside, if you’re lucky, you can end up getting more marks! In the next 2 cases, the expected probability is greater than 0. This means that the rewards are greater than the risks involved, at least in terms of marks. So if you answer a number of such questions, chances are you’d get more marks than if you didn’t make any such guesses. It also means that you would have to be really unlucky to actually lose marks through such guesses. In the case that you have 2 options left to choose from, the expected probability is a very healthy +2 from 2 questions. And intuitively, this makes perfect sense, because while choosing 1 out 2, you have the option of either getting 3 marks or losing 1. That’s a chance you should almost always take – perhaps, just perhaps not if you’re firmly entrenched in Class A. Such cases would also be pretty common during exams. I mean, how many times have you ended up thinking “the answer is one of these 2 choices, it’s either this or that”? Pretty often, right? Well, if you answer 10 such questions in an exam, chances are you’d get 10 marks more than if you had not taken the guesses. Getting any less or any more would require you to get somewhat unlucky or lucky, respectively. 10 marks is typically a LOT of marks in a standard MCQ exam, and could eventually add up to several thousands in terms of a rank.

 

Now let’s consider 3 more marking schemes, as examples:

1) 4 marks for a right answer, -1 for a wrong answer, 1 out of 4 given options is correct – if you guess the answers of 4 such questions, you would be expected to get 1 right and 3 wrong. This gives you an expected probability of +1 from 4 questions answered. This would be what you might call a positive probability exam.

2) 3 marks for a right answer, -1 for a wrong answer, 1 out of 5 given options is correct – if you guess the answers of 5 such questions, you would be expected to get 1 right and 4 wrong. This gives you an expected probability of -1 from 5 questions. This would be an example of a negative probability exam.

Usually in a positive probability exam, you should be looking to answer everything since the rewards outweigh the potential risks. Even if there are questions that you clearly have no idea about, or you are short of time, and don’t have enough time left to even read some of the questions, you should just take some random shots, even without reading the questions and/or options if need be. Chances are that you’ll end up with more marks than you would’ve otherwise.

On the other hand, in a negative probability exam, you should generally be very careful about the risks that you do take. If you take absolutely random guesses, you would likely end up with a score less than what you otherwise would have. Typically, you should be looking to eliminate some options, to get up to a neutral or positive probability before taking guesses. In the second example given above, eliminating 1 option will get you up to a neutral probability. Eliminating 2 or more options will get you up to a positive probability, and now you can take a guess with a greater comfort level.

3) 1 mark for a right answer, no negative markings, 1 out of 4 given options is correct – no negative marking! This falls in a special category of positive probability. Here, the most important point is not the expected probability, but the fact that even in the worst-case scenario, you don’t lose any marks. This means that you should absolutely answer each and every single question, regardless of your class of preparation.

 

Next part – let’s start by taking a small test here. Here are the rules : +3 for a right answer, -1 for a wrong answer. You MUST answer every single question. And please do try your best to score as much as you can:

1)  Who is my favourite tennis player?

a) Juan Monaco

b) Roger Federer

c) Rafael Nadal

d) Steve Darcis

 2) Who is my favourite football player?

a) Cristiano Ronaldo

b) Julian Schuster

c) Pablo Hernandez

d) Lionel Messi

3) What is my favourite TV series?

a) Father Ted

b) Friends

c) The IT Crowd

d) Red Dwarf

4) What is the OS of my phone?

a) Android

b) iOS

c) Windows

d) Symbian

 

Answering complete? Good!

Now before I reveal the answers let’s take a closer look at the questions and the corresponding options. All 4 correspond to some personal detail/preference of mine. So, unless you know me personally, it’s rather unlikely that you know the answers to either of these. However, it’s also very unlikely that you’ll end up with a negative score if you’ve answered the questions properly. Sure, some of you reading this just might, but if you do, let me assure you that you would be in the vast minority.

So, what is it that makes this possible? Let’s look at question 1 first – there are 2 obvious candidates here, Roger Federer and Rafael Nadal, and you would’ve probably picked 1 of these 2 (if not, why? Just… why?). Let’s analyse why – these are 2 of the greatest players of their generations and of all time, both with their own massive fan bases. While the other 2 options are both pretty decent tennis players in their own rights, they definitely don’t have the same appeal as a Federer or Nadal, and are probably only known faces to people who follow tennis rather closely. Maybe you thought of this while selecting your answer. Or maybe you didn’t even need to consciously think of this, your subconscious mind might have done this analysis for you in a split second. Either way, you were eventually left with just 2 realistic options. One lesson here is that you need to trust your intuition or subconscious when it comes to probability. It is always based on some piece of information. So if your intuition says some option is slightly more(or less) likely, you can take it that that option is at least slightly more(or less) probable. Anyway, you are now left with 2 choices. Of course, you cannot be absolutely certain that the 2 discarded options aren’t correct. So in this case, you can assume that if you answer two-point-something such questions where that something is quite small(so two-point-something comes to something like 2.1 or maybe even less), you would get 1 right. So, the expected probability is slightly-less-than-two out of every 2 questions answered. Now, it’s pretty impossible to find the exact probabilities here, but it doesn’t really matter, does it? You can clearly understand that this is a case of positive probability, and by some distance. So you should definitely answer the question (even without the must answer condition). Also, amongst the remaining 2 options, you can try to select the more likely option based on certain heuristics. This would further improve your expected probability. One such heuristic that you could’ve used is the relative popularity of Roger Federer and Rafael Nadal amongst fans and the media. I won’t get into that here, though, for obvious reasons.  Forgive me!

The second question is also very similar in nature to the first one!

Question 3 is a bit different in that instead of 2, here there is just 1 option that immediately stands out. Friends is arguably the most popular TV series of all time, while none of the others come that close to matching its immense popularity!

Question 4 again has 2 primary candidates, Android and iOS. But here, Windows provides a relatively more likely third alternative as compared to the other 3 questions. Therefore, here the expected probability would be slightly lower. However, it’s still clearly positive probability question with some margin to spare.

So, that’s 4 positive probability questions! Why? Because the probability has been taken to the positive spectrum by the application of some logic or heuristic. We’ve not deterministically excluded even 1 option in the above 4.

Well, you can check what you’ve really scored now. Here are the answers to the questions:

1) b) Roger Federer

2) d) Lionel Messi

3) b) Friends

4) a) Android

There are quite a few other things that you could do differently for MCQ exams, which could greatly increase your chances and proficiency in these exams. These include tweaking your methods of preparation, deducing answers using the methods of elimination and backtracking, solving questions using brute force, etc. These will be covered in another article.

Comments

  1. Appreciating the hard work you put into your blog and in depth information you provide. It’s nice to come across a blog every once in a while that isn’t the same out of date rehashed material. Great read! I’ve bookmarked your site and I’m including your RSS feeds to my Google account.

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