A Guide to Negative Marking

Negative marking in Multiple Choice Answer(MCQ) exams is becoming more mainstream in competitive exams all across India. The intent of negative marking is to prevent students from taking guesses at answers. It is typically quite effective because of loss aversion.

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Before we move onto the analysis. Some definitions:

Term	Definition
NumChoices	Number of Choices of answers in an MCQ
NegRatio	The ratio of the number of positive marks one gets for a correct answer to the number of negative marks one gets for an incorrect answer. If it is a +4 and -1 exam, the ratio is 4. If it is a +1 and -2 exam, the ratio is 0.5
Random Strategy	Strategy to mark the answers not known, at random
Refrain Strategy	Strategy to refrain from guessing, if the answer is unknown

Analysis Level 1:

On the top level, the probabilities seems simple. For example:

If there is an exam with the NumChoices as 4 and NegRatio as 4, the the Math works like this:

Probability of getting answer correct (Pc)	=	0.25
Probability of getting answer incorrect (Pi)	=	0.75
Marks probabilistically obtained	=	(Pc * correct answer marks) + ( Pi * incorrect answer marks)
	=	(0.25 *4) + (0.75*-1)
	=	0.25

In fact, for different NumChoices and Neg Ratio this table can be used to predict marks obtained by Random Strategy at average, given large number of attempts.

	NegRatio ->	0.5	1	2	3	4	5
NumChoice	2	-0.25	0.00	0.50	1.00	1.50	2.00
	3	-0.50	-0.33	0.00	0.33	0.67	1.00
	4	-0.63	-0.50	-0.25	0.00	0.25	0.50
	5	-0.70	-0.60	-0.40	-0.20	0.00	0.20
	6	-0.75	-0.67	-0.50	-0.33	-0.17	0.00

This analysis however does not tell one too much about what is the right strategy, because it doesn’t tell you how often does the guessing strategy fail you. To better understand we need the Lottery Analogy.

Lottery Analogy Imagine there is a lottery with a 1 in a billion chance to win ₹6 billion. The lottery ticket costs ₹5. By the probability analogy just established, the lottery ticket is worth ₹6 billion / 1 billion = ₹6. So therefore by above practice, it makes sense to always buy the ticket. But for a poor man, whose livelihood depends on the ₹5, it would be stupid to invest in the lottery because there is 99.999999% chance he will starve.

Similarly, if you are in a competitive exam with only one shot, how do you as a student decide, if this one shot is worth the risk. Or what is a probability of getting a zero or greater score by the random strategy?

For the analysis so far - only in the unique situation where the probabilistic result is zero, is there clarity over there being 50% chance of the Random Strategy being better than the Refrain Strategy.

Analysis Level 2:

This time we will have to move away from direct probability and move to a more brute force solution. For starters, we will simulate someone taking the tests with 100 questions being answered at random from 4 choices with +4 for a correct answer and -1 for an incorrect answer 1,00,000 times.

The scores from the simulation look like this -

The above plot indicates the % of tests in the simulation in which the score was higher than the corresponding number indicated on horizontal axis.

So, interestingly for this case only 10% of the time does one end up with a net negative score on randomly guessed questions using the Random strategy. On the other hand there is the same 10% chance to get 57 free marks. The average as predicted by the table above is around 25 marks. So Random Strategy could be very beneficial.

Here is the same plot for +3 and  -1 type of exam. The gaps between the columns are because in this scenario there are some score that are not possible.

As expected from the table above there is close to 50% chance of the Random Strategy doing better than the Refrain Strategy.

This gets a little bit worse for +2 and -1

Analysis Level 3:

One is not always guessing

There is the possibility of eliminating at least one answer. For the +4 and -1 case the plot shifts to this

Significantly, not only has the average moved to 70, but also the chance of random strategy failing you is now down to 0.1% or 1 in a thousand.

On the test side, there is also the chance that the exam does not have randomised options. That is the options A, B, C and D are in the order the question setter wrote them. Typically, a test has options like "All of the above" or "None of the above" are not randomised. In these situations, the probability of the following option being the correct answer is this:

A	B	C	D
20%	40%	30%	10%

If the simulations are run with this in mind and the answer entered is always B, then the distribution for the Random Strategy looks like this-

Now no test in the simulation scored < 0. In fact the average positive gain has moves to 127 marks.

Finally, typically there is a hunch that one might have, let's say that means that the probability of the hunch being correct is 50%. In that situation the distribution shift looks like this -

No test in the simulations with score < 0 and Average moves upto 152.

Conclusion:

For negative marking exams with NumChoice = 4 and NegRatio > 3, Random Strategy is looking strong. Specifically, in situations where the test is non-randomised, an option is eliminated or there is a strong hunch, there is almost no possible way to be negatively affected by the Random Strategy.

Things you must consider before you do the Fast and the Furious jump

Fast and the Furious 7 comes out this week and we are very intrigued by this trailer which has our heroes jump out of a plane in cars.

You have to love how the guys in the Fast and Furious 7 find innovative ways to get their cars to weird locations. Were there other ways? Yes, but this is cool isn’t it?

If you decide to make a similar drop for yourself, may be to crash a party in style or drop straight into a major sporting event here are the top 5 things you must check for.

1)    Get your Parachute pick up point correct

It is critical for you to get the place you latch the parachute onto your car correct. As cars are designed to run on four wheels it important that it lands on its four wheels so that you can saunter off to do what you want once you have landed. This can be easily established by working out where the Center of Gravity (CG) of the car is, and the chute pick up point must be directly above it. A convertible will probably not work.

2)    Get your parachute size correct

It is important you have a soft landing not wrecking your suspension. Road cars can be modified not too much to continue operation post approximately a 1 meter jump. This gives the car a landing speed of 6.26 meter/sec.

The parachute must be at least 550 square meters per ton of the vehicle and driver weight. The diameter of the chute for a 1000kg (1 ton) vehicle must be at least 26.5 meters.

3)    Reinforce your roof pick-up point

The roof of a car is not designed to be connected to a parachute ^{[citation needed]}. It would be a good recommendation to add a roll cage to your car and attach the chute pick up on the top of that to avoid this situation –

4)    Make sure you have a system to steer the parachute

A car is useless if it lands on the top of a tree or in the middle of a lake. Parachutes can be steered by steering toggles. Make sure you have them inside the car before you set off from the plane. Even better if you can connect the toggles to the steering rack.

5)    Make sure you check your oil levels, tire pressure, coolant level and fuel

After you have done all of this, you don’t want your car to break down because you did not have brake fluid in your cylinders, right? So please check these basic things before you jump.

India Vs New Zealand - Second best team at the world cup?

The Cricket World Cup 2015 is finally over and Australia are the champions again. They are certainly the best team in this competition. In fact, knock out tournaments are well designed to find out the best teams in the competition. In fact, it can be shown that a knockout system is the most efficient method to find out which team is the best in a given set assuming the results are transitive.

This format unfortunately is not a good one to find the second best team in a set of teams. For example Australia beat Pakistan, India and New Zealand in that order to win the World Cup. Which means that any one of the 3 teams could have been the second best team after Australia and were just unfortunate to play Australia early in the tournament. However, there is more data from this world cup to play with.

The elaborate round robin group stage before the knock out stages provides more results to play with and apply transitivity. For example, famously India played Pakistan on the second day of the World Cup and India won that encounter. Which means that one of New Zealand or India is the second best team. Unfortunately, since the both India and New Zealand were in different groups and they won all their matches in group stages, data from the group stages does not sort the problem of which is the second best team.

In fact if we try and rank the teams in the knock out stages using world cup data and transitivity this is the only unresolved ranking:

Note: The game where New Zealand beat Australia is ignored because Australia beat them in a game that mattered more.

If New Zealand would have beaten Australia in the final then the transitivity could have accounted for the full ranking. However, now an India Vs New Zealand match is needed to sort this once it for all. This will make the world cup more productive.

Why do couples at Marine Drive sit equally spaced from one another?

An interesting observation made by @TheToothsayer when we recently visited Marine drive was that the couples that sit at the sea front, sit equidistant to one another, quite precisely equidistant that too and over large distances. This nonchalant observation has more interesting insights than what is prevalent on the first look.

Photo courtesy: Kunal Bhatia - www.mindlessmumbai.com

Two odd things – firstly, there is no clear coordinator imposing that they should be sitting at a certain distance from one another and secondly, there seems to be no conscious effort to do this coordination. It just happens, it happens every single day.

How this formation happens is perhaps not too difficult to imagine if we piece this together in the chronology of the couples arriving at Marine Drive. Much like the XKCD Urinal Protocal Situation, we can postulate 3 basic rules that a logical couple would follow to find themselves a spot at the sea front -

1. The couples like privacy ^{[citation experience needed]} and they will optimize their distance from other couple such that they will maximize the minimum distance they have from any other couple.
2. There is a distance after which increasing the minimum distance from the next couple has no added value. Simpler way to state this is that a couple once a certain distance away from the next couple will just choose that spot because it’s not worth walking further.
3. Similarly, there is a distance that is the minimum acceptable distance from the next couple, that a couple can tolerate. This means that if the maximum minimum distance from any couple is less than this distance, then the couple will choose to wait for a spot over squeezing themselves awkwardly between the other couples.

Based on these the situation pans out to be something like this –

What is remarkable here is that all the 3 rules were for each individual couple and there was no coordinating agent that ensured that an equidistant pattern was formed. This is a purely self-organized pattern, which is a social trait of human beings. In fact, it would be safe to say that if there had been an established authority that made couples sit equidistant, then there would be resistance and it would not work with the smoothness it does naturally.

There is one situation when this logic does break down. When two neighbouring couples leave at the same time, where would the arriving couple choose their spot?

If the 3 logical rules are to be followed, then position A is the logical choice but Position B and C would keep the couples equidistant. I asked 30 random people on the train I am on right now, which position they would choose (I told them what I was doing after they chose). 27 out of the 30 chose position B. I would have chosen Position A because I would like to think I like logic over symmetry; however that is probably my frame of mind when writing this. Nature loves symmetry and it is clearly manifested in the human mind.

And THAT is why couples at Marine Drive are equidistant to one another.

Edit: This article has resulted in an overnight Phd into co-operative behaviour in the natural world – Seems like this is not only a human trait; interestingly here is a picture of ants self-organizing into equidistant pattern around their food.

Also surprisingly, Cognitive Biology is way simpler to make sense of over Photometry.

The Lunar Program - A Reflection

This is one of the most depressing plots I know -

Getting to the moon was significantly difficult to do in the 1960s than it is today, but still for some reason we don't go there. In the 1960s and the 1970s there was a cold war that caused enough motivation for J.F. Kennedy to pile up the country’s resources outside the NASA headquarters to get to the moon at any cost. Today, there is no cold war and economics has dominated the decisions to end the shuttle program. We need economic demand to go to the moon and we need to do this fast. I am going to do a few blog posts about a few ideas I have thought of. Consider these as Elon Musk style open patents, from someone who is significantly more lazier than him. We (humanity) must get some engineers out there to solve our problems.

What the world needs right now is a solution to the energy crisis. The Moon probably does not have oil, considering oil was formed from once fossilised life-forms. The Moon probably does not have coal or natural gas either for the same reason. What the moon has is a lot of barren land. What use is a lot of barren land that is visible from most of the Earth in the night time?

Meanwhile, the FIFA world cup has just concluded and there has been a lot of talk in Brazil, over whether it was worth hosting it. Sports are good. They are fun, bring the world together in friendly rivalry, boost local economies, provide great conversation starters and are a welcome distraction / entertainment source to the daily life. All of this comes at a cost and it is a good idea to bring these costs down. I am sure the FIFA world cup is expensive to host, one of the most visual expenses are the stadiums. The stadiums are large and extravagant. All of them now have flood lights which turn night to day at the stadium. This obviously is expensive to do, but night matches are economically worth it. It makes the game happen at prime time, and an artificial lights game is quite a spectacle.

It would be nice if we could play sport in the moon light, but unfortunately moon light is not enough.

This is an engineering blog so “not enough” is not an acceptable answer. Illumination can be measured. The SI unit of the Illuminance is ‘lux’. This is what light meters measure. When a cricket umpire holds out a meter, he is measuring the light per square meter. The SI unit of light is candela, which is the amount of light provided by one candle (a definition, a bit like horse power, I am amused by the idea that there is a standard candle/horse somewhere in the world which was used as reference). For cricket ‘lights’ is offered to the batsman if the illuminance is less than a 1,000 lux. However, the FIFA regulations are tougher, demanding at least an average of 3,500 lux for an International televised game. Moon light at an average is 0.4 lux. So moon light is “not enough” by 17,500 times.

Why is the moon light so low? Moon light is essentially Sunlight that is reflected by the Moon. The moon is at an average equally far away from the Sun as the Earth is, so the illumination received by the moon is the same as Earth on a sunny cloudless day, which is 1,20,000 lux. On a full moon night, the Earth receives a small fraction of light reflected by the moon, as this diagram will explain (Note the log scale).

The reflection co-efficient of the moon is 0.136 which means that 13.6% of the light incident on the moon is reflected. The fraction of the reflected light that the Earth would intersect is -

This value is ~.7 lux but the average illumination of the moon light measured on a clear night is 0.25, which means there is a 35.5% efficiency factor in there which accounts for light lost in the atmosphere (due to lack of clear skies) and averaging out of the fact that there is an angle of incidence involved both on Earth and on the moon.

To have moonlight soccer, we need to start working on making this more efficient. Let’s start with the reflection co-efficient - place mirrors on the moon. We clearly need to go to the moon to do this. We have managed to produce 99.99% reflecting surfaces but let’s cover the moon by 99% reflecting mirrors. However, even if we cover all of the moon, our new ‘Disco ball’ moon will only reflect ~7.2 times more light which is still nowhere enough.

Only a fraction of the reflected light actually hits the Earth, so the rest of the reflected light is wasted. We must have movable ‘sunflower mirrors’ that always point in the direction such that it reflects light to Earth. These movable mirrors will directly impact the light spread related efficiency factor. In fact, since all the light reflected by it is directed towards the Earth this efficiency factor can go up very close to 1. If the entire Moon surface is used to illuminate the entire Earth's night sky on a full moon day, the average illumination would be -

Adding the 35.5% efficiency factor accounting for atmosphere loss and angle of incidences, the average illumination of the Earth at night would be 3090 lux which is enough to host a cricket match anywhere on Earth any time of the day.

However, we still have all of the moon covered with reflectors that illuminate all of the Earth. This is still wasteful and very confusing to wild life.  We can consider reducing the effective reflector size. To illuminate the all of the Earth with 3090 lux we need the entire moon surface. To illuminate only a specified area on Earth with a desired illumination, the reflector area on Earth can be calculated by -

For a cricket field to be illuminated by 1000 lux, we need reflector area approximately 30 times smaller than a cricket field. For a FIFA match we can do with the total reflective area of 10 times smaller than a football field. They seem manageable, don’t they?

These reflectors can change their angle of incidence, which means the same reflectors can light up a football match in Australia, Europe and South America on a single day. Stadiums simply rent the reflectors on a need basis and pay by the hour. Beyond sport, this gives us the power of shining light on any part of Earth whenever necessary. Imagine how useful will it be to literally light up a rescue operation if a train were to have an accident at a remote location in the middle of the night. To light up 1 km of a 6 lane highway as per this, we need only ~35 meter square of reflectors.

Anyhow,  I'm looking forward to sporting events scheduled as per the lunar calendar, and a cricket match being interrupted by a lunar eclipse.

Hash-tag -> #GoToTheMoonAgain

P.S.: I also learnt that Photometry as a subject has one of the highest (Difficulty to understand / How simple it looks) ratio that I have come across. Also, solid angles are evil.