Scientific Guessing

Mathematics is often considered an exact science.

2+2 is always 4. cos²x + sin²x is always 1.

However, this is not always the case. Mathematics also deals with probability - a deterministic science.

Why Guess?

Objective examinations are considered a test of analytical skills and a test of nerves.

Now, it also tests your knowledge of probability. (Didn't it always test that?)

Regardless of the syllabus, objective pattern exams will test how well you apply probability.

Surprised?

Consider this. JEE in 2008 had 132 questions. In these, I would reckon there were 20 questions where I had a doubt. I wasn't sure. It was 50-50.

Did I guess?

Of course, I did.

Guessing in JEE

The objective questions had a +4/-1 marking pattern (+4 if you get it right, -1 if you get it wrong).

I figured out - even if I got 4 questions right out of 20, I wouldn't be losing out. (4x4 - 16x1 = 0).

How much probable is that?

Probability of getting exactly no question right = ²⁰C₀(1/2)²⁰

Probability of getting exactly one question right = ²⁰C₁(1/2)²⁰

Probability of getting exactly two questions right = ²⁰C₂(1/2)²⁰
Probability of getting exactly three questions right = ²⁰C₃(1/2)²⁰

Probability of getting exactly three questions right = (1/2)²⁰ { ²⁰C₀ + ... + ²⁰C₃} = (1/2)²⁰ (1351) = 1.28 x 10^-3

Probability of getting at least 4 right = 1 - 1.28 x 10^-3 ≈ 1

Once I figured this out, my JEE paper was practically finished.

I guessed left-right and finished with a rank of 1594.

Double Check

Still cynical?

Feel that something somewhere is wrong.

Let us double check.

Probability of getting either none right or one right or... 20 right = (1/2)²⁰ { ²⁰C₀ + ... + ²⁰C₂₀}

How much is ²⁰C₀+²⁰C₁+...+²⁰C₁₉+²⁰C₂₀?

It is 2²⁰!

We then get the probability = 2²⁰ / 2²⁰ = 1

That is what we should be getting!

What if I had no idea and simply guessed?

Probability of getting exactly

none right would then become ²⁰C₀(3/4)²⁰

one right=²⁰C₁(3/4)¹⁹(1/4)¹

two right = ²⁰C₂(3/4)¹⁸(1/4)²

three right = ²⁰C₃(3/4)¹⁷(1/4)³

Probability of getting less than 3 right = (3/4)¹⁷ { ²⁰C₀(3/4)³ + ²⁰C₁(3/4)² (1/4)¹ + ²⁰C₂(3/4)¹ (1/4)² + ²⁰C₃(1/4)³ } = 0.225

Probability of getting at least 4 right now becomes 1 - 0.225 = 0.775

It isn't as safe to guess now, even though the probability is still in the candidate's favour.

AIEEE/BITSAT etc.

AIEEE has the +3/-1 marking scheme.

If you use the same method to calculate, you'll find that

the probability of scoring non-negative in case of 50-50 ≈ 1

when one has no idea = 0.61

BITSAT has bonus questions if you complete all the questions. Suppose you have 20 questions left out. Should you guess and go for the bonus questions?

You are now empowered to decide for yourselves!

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Mathematical PJs

The Day Before The Exam

[This is the text of the post that I wrote on JEE-nius the day before the JEE - comma, full-stop - no changes made. However, this could well be applicable to any other exam.]

I spent 3 years preparing for JEE and cracked it not once but twice. The second time - I secured a rank that was one-third my first attempt rank with one-third the preparation in my first attempt.

So how did I do it? What was the secret ingredient that was missing in my first attempt that got the magic going in the second?

Mindset

More than a test of analytical abilities, JEE is a test of nerves.

The D-Day could be anybody's day.
It could be your day.

You have to BELIEVE that it is going to be your day. Once you believe that it is going to be your day, there is no force in this world that can stop you from cracking the JEE and coming out with flying colours.

Being Alert

Get into an alert frame of mind. You wouldn't want to come out of the examinations regretting that you failed to read the instructions. Or that you could have solved the problem you didn't solve.

Go out for a Walk

Whenever someone asks me - What did you do on the day before the exam? I tell them this - I went for a walk with my father.

Avoid INACTIVITY and HYPERACTIVITY. They only increase anxiety. The best way to stay cool, calm and composed is to go out for a walk. It can do wonders tomorrow.

Avoid Caffeine

Caffeine increases the alertness levels. But it also increases anxiety.

Avoid coffee, cold drinks or anything that has high caffeine content.

Have normal food. Nothing extravagant. You wish to be in your mental and physical best - extravagant food will not help.

Prepare Well

I am not asking you take out your notes and start revising. That part of your preparation is over.

Do what remains. Get your pen, pencil and eraser ready. Have backups. Check your admit card. Be sure of the examination venue. Start early tomorrow.

Have a good sleep

Do not try to wake up early. No use revising now. Having a good sleep will help you maintain an alert state of mind tomorrow.

Tomorrow

Do not fret. You know it is going to be your day. The Day of (Your Name). Reach the examination venue well before time. Go and check your room no and floor.

In the examination hall, follow the instructions of the invigilators. Read the instructions CAREFULLY before starting.

And remember, if you cannot solve a question - even Chintu/Montu cannot. It is all about maximizing your marks. No one gets all the questions right.

Tomorrow is going to be your day. You are going to rock JEE.

All the best.

[P.S.: The post started appearing in sites such as GoIIT hours after it had been posted on JEE-nius. I would request people to refrain from copying the posts. They can, however if they wish, provide the link to the post to whoever is interested. The content is copyrighted]

What it takes to crack JEE

If you are in the age group of 16 to 19 years and an Indian national, chances are that you might be preparing for either Engineering or Medical Entrance Exams. Among those exams, I guess JEE is the one which is well - let us say hyped up a lot. Till two years back, it was supposedly the most difficult exam to crack. At least now, it isn't. So what does it take to crack JEE. Let us list them down.

Intelligence?

Preparation? (aka Coaching)

Confidence?

Family Support?

What is it that separates a fellow who has cracked JEE & one who hasn't. Having cracked JEE not once but twice ;) I can say with confidence that you must add one more factor to this list

Luck

This is an often overlooked factor when an aspirant goes to give the JEE. No matter how good your preparation may be, no matter how much ever intelligent you may be - I think no IITian can deny having luck on his side on the day of the JEE. He may have made his share of silly mistakes - that everyone does. But without good luck, (s)he would not be in IIT.

You have to be in your best frame of mind & in good mental as well as physical health. If the question paper has more questions from a topic which you are very good at, you have luck on your side. On the other hand if the questions are from the topic you hate the most, well then - Better Luck Next Time(BLNT)

Then comes

Intuition

In the limited time of 6 hours one has to solve a mammoth 130 odd questions, which is very exhausting mentally. Here comes the role of intuition. Leaving out certain sections due to lack of time, preparing topics that by chance is asked thoroughly in the paper & a few odd guesses here and there it is all intuition.

Confidence? Important but not imperative. I guess luck & intuition automatically result in a confident aspirant. If you don't have them on your side, then as I said BLNT.

Family Support? Well if it is about letting you do a late night or leaving you till the exam centre on the day of the exam - lots of books & all, I guess it really counts. But if its about pressuring you to give JEE for the heck of it - I guess it doesn't help much.

Intelligence? Nah, JEE is very much trainable. The very intelligent may be (rotting away?) outside IITs. All you need is some luck & a little a bit of intuition. Coaching, intelligence & preparation are all secondary

Best of LUCK!