Tuesday, April 19, 2011

Tera Kya Hoga Kalia? - Solutions

The solutions to the problems posted on 23 March.

Thanks Mark and Abe for posting your solutions. You got the first six questions right.

The entire problem including the questions with all three levels included is given here as the Solution.

An exercise in probability theory

You are of course familiar with the Kithne Aadhmi The scene in Sholay.  For those of you who are not, this is how it goes.

Kalia and two others (they are not named in the film - let’s call them Amar and Akbar) are sent by Gabbar on a special assignment to liquidate Jay and Veeru. The three brigands are thrashed by our heroes and return empty handed.  Gabbar is not amused.  He lines them up in front of him and starts loading his revolver.

It’s the famous Hindi film revolver with six chambers.  Gabbar loads three bullets in three adjacent chambers.  He rotates the chambers effectively randomizing them. He turns and faces Amar, Akbar and Kalia who are standing in that order respectively.

The plan is simple.  He will point the gun at Amar’s head and pull the trigger.  He will then proceed to Akbar and do the same.  Kalia is the last in line, and he will do the same to Kalia.  He will not randomize the chambers between shots. You are now at the point where he is about to start this famous Bollywood version of Russian roulette.  You are part of Gabbar’s gang and you are also the bookie.  You would like to offer odds to the rest of the gang.  Before that, however, you need to calculate the odds.  So you take out your pad and proceed to calculate probabilities for the following possibilities.  You had better be right, since your money is on the line.

(Let us assume the three bullets are loaded in Chambers 1,2, and 3. Chambers 4,5, and 6 are empty – Gabbar will start at any chamber (with equal probability) and shoot consecutively (without randomizing in between shots) at Amar, Akbar and Kalia respectively)

    1. Before the shooting starts: What is the probability of Kalia dying? (1/2)

    2)      After the first shot: if Amar is dead, what is the probability of Kalia dying? (Since Amar is dead, he died from one of Chambers 1,2 or 3. The next shot will thus be 2,3 or 4. And the shot after that meant for Kalia will be 3,4 or 5. 3 has a bullet, 4 and 5 don’t. Hence 1/3)

    3)      After the second shot: If Akbar is dead, what is the probability of Kalia dying? (you don’t know Amar’s fate) (Akbar was shot by either 1,2 or 3. Kalia will have to deal with 2,3 or 4. 2 and 3 have bullets in them; 4 is empty; hence 2/3)

    4)      After the second shot: If Amar and Akbar are both dead, what is the probability of Kalia dying? (Amar and Akbar were shot by “1 and 2” or “2 and 3”. So Kalia will get either 3 or 4; 3 has a bullet, 4 does not. Hence ½)

You also know that Gabbar is an avid gambler and likes to play cat and mouse with his victims.  He has already told you his alternate game plan. After the first shot, Gabbar might give an option to Akbar to choose whether he wants Gabbar to randomize the chambers or not. In such a case:

    5)      In case Amar is dead, what would Akbar choose? (Amar got hit with 1,2, or 3. Akbar will get 2,3, or 4 which means 2/3 probability of dying. Whereas if he randomizes, there are 2 bullets left in six chambers – chance of getting killed in 1/3. So he will choose to Randomize)

    6)      In case Amar is alive, what would Akbar choose? (Kalia will get one of 5,6 or 1 – chance of getting killed is 1/3 if he does not Randomize. On randomizing the chances of getting killed are half since there are still three bullets left. Hence he will choose not to Randomize)

Akbar happens to have an M.Sc. in statistics and hence is likely to be very logical in his choice.  He also wants to live if he can help it.

Now that you have calculated all the above, imagine another scenario, where Gabbar is going to randomize after every shot.  In such a scenario:

(Please refer to the excel file in the link attached – the solutions below refer to it

    7)      Before the shooting starts: What is the probability of Kalia dying?

    [Outcomes 1+3+5+7 = 25/72]

    8)      After the first shot: if Amar is dead, what is the probability of Kalia dying?

      [Amar dies in outcomes 1 to 4, of which Kalia dies in outcomes 1 and 3. Hence (2 + 8)/(2+8+10+16) = 10/36]

    9)      After the second shot: If Akbar is dead, what is the probability of Kalia dying? (you don’t know Amar’s fate)

      [Akbar is dead in outcomes 1,2,5 and 6. Of which Kalia dies in 1 and 5. Hence, (2+6)/(2+10+6+12) = 8/30]

    10)     After the second shot: If Amar and Akbar are both dead, what is the probability of Kalia dying? [1/6]

Wednesday, April 6, 2011

Self Delusion

Now that the Indian team has won the World Cup, there will be a spate of articles trying to analyze the victory. But before getting into that, we need to savor the victory, don't we? The whole nation is into a self-congratulatory mode, exulting wildly at the win. The players are being wined, dined, feted, hoisted, hosted, and generally smothered with attention. Every word that they utter is noted down for posterity. The fact that posterity lasts for a very short while in the modern age is something that we should not think too much about.


I have nothing against the victory. I think the players have done a great job and we need to be proud of them. But stepping back a little bit, there seems to be a little bit of perspective required somewhere.


All articles and interviews aim to get some insight into what got the team where they are today. What is the secret of their success?  And this is where it becomes very interesting. From whatever I am able to gather, the secrets of success as divulged by various members of the winning team are: Sachin Tendulkar, hard work, Guru Dera Wala Baba, Sachin Tendulkar, commitment and focus, dedication, planned and targeted preparation, Sachin Tendulkar, an excellent coach, hunger for victory, discipline, Sachin Tendulkar…


If the stock market goes up by 700 points today, the newspaper headlines say that it is due to a stable situation in the US, less than expected bad news from Japan's nuclear disaster, positive outlook for the Indian economy, stable government at the Centre, etc. If on the other hand the market goes down by 700 points, the newspaper headlines will say that it is due to worsening inflationary outlook, bad news from Japan's nuclear disaster, price of oil, situation in Libya, increase in commodity prices, etc. Depending on what is the event that happens, the right set of reasons is chosen.


The analysis of the cricket team's victory also seems something like that. We know that victory requires one part each of hard work, commitment and focus, dedication, preparation, coaching, hunger, discipline, and five parts of Sachin Tendulkar. Why did we need to play the entire tournament to know that? And how come these conclusions are being touted as great insights?


No one bothers to ask why it is that the other teams lost when they too had all the above, perhaps more than one part each in some cases. (Actually, I know the answer to that one – they didn't have Sachin Tendulkar!). Or why we got knocked out at the league phase the last time around, when we had all the above ingredients in equal measure?


Not just in cricket or the stock markets, but in every area of life, we tend to justify our views and biases, based on analysis carried out on hindsight. Only those things are allowed to insert themselves into reckoning that fit in with our preconceived notion of the world. The results from the analysis in turn bolsters our conviction in our own arguments. Everyone thinks alike and there is immense pressure, both overt and subtle, to agree to what is the obvious truth. The more we analyze in this fashion, the more we are convinced that we are right, and the more self righteous we become. Cricket and stock markets are just examples; the malaise is spread much wider.


And we happily wallow in self delusion. It's not a bad way to be. At least we are happy!

Monday, April 4, 2011

Captain Cool

The whole nation accepts him as leader,
Perhaps as one who was born to lead.
He rose up from the small towns of nowhere
to captain the Indian cricket team.

He heads a team that is full of titans,
with egos fed on adulation.
To them, he is first among equals,
and they acknowledge him as their captain.

He is bold, but not seen to be reckless.
He is cautious, but surely not feckless.
He is rustic, in a charming way,
and suave, at home in any ambience.

He seems to hold his head under fire,
not carried away with every current.
In stormy seas, he has his own anchor,
which seems to moor him, and give him strength.

No one in India is acknowledged,
as completely as undisputed.
What is his appeal that they hold him
high and shower him with their esteem?

If it's hopes of a nation he carries,
he does not let the burden show.
Just watch him, taking to the shadows,
after every brief moment of triumph.

Dinesh Gopalan
4 April, 2011

Friday, April 1, 2011

The Tournament Finals!

Excitement risen to fever pitch,
Eagerly goading each other on,
The nation waits with bated breath,
For the tournament just anon.

To each it's like his own fervent wish,
what, this moment, he wants the most;
Multiply that by a billion,
number of hearts stilling with hope.

Why should one event mean so much,
Catharsis to an entire nation?
Can't we treat it just as a game,
Stripped of such high expectations?

Or is it that we need these rituals
of bonding in collective hope?
Moments of victory and defeat
that we all agree mean much more?

A coming together, a bonding,
eager awaiting of the outcome,
where everyone comes together as one,
and for once, prays to the same Gods?

Dinesh Gopalan
1 April, 2011