I’m bemused when I see today’s exams. Many are just multiple choice tests. There’s a question like “Who said ‘Never was so much owed by so many to so few’?” and you have to guess who said it: Franklin Roosevelt, Mahatma Gandhi, Adolf Hitler or Winston Churchill?

Of course I’m exaggerating. Even multiple choice tests can be complex! For example, you might have to answer 100 such questions in 30 minutes; so there’s time pressure. Or there could be negative marks; so you have to be circumspect.

There’s also a philosophical sort of objection: if the answer picked is wrong, you know for sure that the candidate doesn’t know the answer. But if the answer picked is right, can you be sure that the candidate truly knows the correct answer? Could it just be a lucky guess?

That’s the crux of the matter. How do we conquer the scourge of guessing? How do we ensure that we select a winner, and not a gambler?

A candidate once explained his strategy rather well: “I ask if there are negative marks. If there are no negative marks, then I first attempt all the questions for which I know the answer … and then I randomly guess all the remaining answers!”

That’s a smart thing to do. Imagine that there are 100 questions — with four choices — each carrying 1 mark. Imagine that there are two candidates … let us call them Hemant Satyavadi and Rohan Khiladi. Imagine that both Hemant and Rohan know only 40 answers. Hemant attempts only 40 questions and leaves. What’s the highest mark Hemant can get? 40/100! Rohan however reacts differently: he first ticks off the 40 answers he knows, and then randomly guesses the other 60 answers. If Rohan is very very very lucky he could get 100/100, but even if he is very very very unlucky he’ll still get 40/100. On the average expect Rohan to get 25% (one out of four) of his guesses right; so he should get about 15 marks more to end up with 55/100.

Now that’s unfair! Both Hemant and Rohan deserved to get 40/100. If 50/100 was the qualifying mark, both should have been out. Instead we now have a situation in which Hemant is out and Rohan is very probably in.

The preferred way to discourage guessing is to introduce negative marks. So what sort of a negative mark would be right? How about -1 for a wrong answer?

A moment’s reflection will show that this is a very stiff penalty. Even if the candidate has the slightest doubt, he will choose not to attempt an answer, because every wrong guess will ‘undo’ a correct answer.

So what about a lower negative penalty? What about -1/4 for a wrong answer? It is easy to see that -1/4 may not be a sufficient deterrent to prevent guessing. Imagine that Hemant and Rohan don’t know answers for any of the 100 questions. Hemant — who never guesses — will leave with 0/100. Rohan — who always guesses — will, on the average, get 25 guesses right and 75 wrong. The 25 correct guesses will fetch him 25 marks but the 75 wrong guesses will now take away 75/4 = 18.75 marks. So Hemant will end up with 25 – 18.75 = 6.25/100 when he too deserved 0/100! Some simple arithmetic will now show that if we give -1/3 for a wrong answer both Hemant and Rohan get the 0/100 that they truly deserve.

So to summarize: In a multiple choice test with four options, always guess if there are no negative marks; continue to guess till the negative mark drops to -1/3; and don’t guess if the negative mark falls below -1/3. If the negative mark is exactly -1/3, it doesn’t matter what you do.

So to summarize: In a multiple choice test with four options, always guess if there are no negative marks; [b]continue to guess till the negative mark drops to -1/3; and don’t guess if the negative mark falls below -1/3. If the negative mark is exactly -1/3, it doesn’t matter what you do.[/b]

I think this is where knowing one’s place is important. If Hemant and Rohan knew the answers to 40 questions, it is important to estimate what the cutoff is. If the cutoff is likely to be 50 or more, they should definitely try guess even if the negative marking is stricter than 1/3. If on the other hand the cutoff is likely to 38 or 40, then even if the penalty is 1/4, one should not guess.

Another place where this same business is important is in limited overs cricket. The team batting first needs to know what is a good score, by `reading the pitch’. Very often, this is badly estimated, and teams lose the chance to make a good score by taking risks necessary for a score in excess of a good score.

Since Rohan has figured out that it pays to guess, he deserves a higher score, don’t you think?

I guess yes. Actually there’s another reason why guessing makes more sense. Even an idiot is often able to eliminate at least one of the four choices. So the probability of guessing right is at least 33%, rather than 25%. If Rohan guesses smart, he could even get 20 guesses right out of 60!

“Rohan — who always guesses — will, on the average, get 25 guesses right and 75 wrong. ”

Can you explain how ?…

probability of getting 25 questions right out of 100 is low given that the probability of getting each question right is only 25%?

Rohan, I don’t understand your question! If there are four options, one out of four random guesses will, on the average, be right and three out of four will be wrong.

Let us say there are 4 questions in an exam.Probability of each question being wrong is 3/4 and right 1/4.

probability of all questions being wrong = (3/4)^4

probability of at least one question being right = 1-(3/4)^4= 0.68

I am thinking on these lines….

I don’t understand what do you mean by “on an average”? Rather than saying that on an avg you get 25 out of 100 right shouldn’t you be calculating the probability of getting at least 25 right?

If you guess randomly every choice has a p=1/4 of being right. I have to say ‘on the average’ because a particular instance may give you 28 right out of 100 or 23 out of 100. But if you average over these individual instances you’ll get 25.