## Wednesday, March 23, 2011

### Tera Kya Hoga Kalia?

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.

1. Before the shooting starts: What is the probability of Kalia dying?
2. After the first shot: if Amar is dead, what is the probability of Kalia dying?
3. After the second shot: If Akbar is dead, what is the probability of Kalia dying? (you don’t know Amar’s fate)
4. After the second shot: If Amar and Akbar are both dead, what is the probability of Kalia dying?

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?
6)      In case Amar is alive, what would Akbar choose?

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:

7)      Before the shooting starts: What is the probability of Kalia dying?
8)      After the first shot: if Amar is dead, what is the probability of Kalia dying?
9)      After the second shot: If Akbar is dead, what is the probability of Kalia dying? (you don’t know Amar’s fate)
10)     After the second shot: If Amar and Akbar are both dead, what is the probability of Kalia dying?

(don’t try googling – you won’t find the answers since I just made all this up)

santhanam said...

Dear Sir
I received this problem thro' my friend Mr.Balu of Coimbatore. I do not know Statistics. However with my little knowledge I have given answers for the 10 questions. I hope atleast some 4 or 5 may be correct. I am waiting for the correct answers.

1. 3
2. 1
3. 2
4. 1
5. Akbar will choose for randamization of chambers because there are 4 possibilities for escape.
6. Akbar will again choose for randamization of chambers because there are 3 possibilities for escape.
7. 3
8. 2
9. 2
10. 1

With regards
JS

Hari Haran said...

DG,

I have no pretensions about my statistical abilities so wouldnt even try. Would look forward to your solution in a week's time.

I'd only say that if only this note was written before Ram Gopal Verma tried his version of Sholay with 'Aag', the nation would have been spared the ordeal. :-))

Abe said...

Hari's humble subordinate here:
1) 50%
2) 33.34%
3) 66.67%
4) 50%
5) Randomize
6) Dont randomize
The last 4 required heavy thinking.
With the crazy workload on me I wouldn't want my boss to think I was slacking off, hence leaving them for better minds :)
Regards,
Abe

Unknown said...

Hi Dinesh, here's my shot at it:

1. 50%
2. 33%
3. 66%
4. 50%
5. Ramdomise - since the probability of life goes up
6. Don't ramdomise - for exactly opposite reasons from above
7. 50%
8. 50%
9. If Amar lives: 33%, If Amar dies: 16%
10. 16%

I have ignored decimals in the percentages. Thanks for the teaser!

Anonymous said...

Great way to inculcate peoples interest in these wonderful subjects.

Jackie Jaishankar said...

Hi Dinesh, got to this from your LinkedIn page. Very interesting - just from a titling perspective, felt "Tera Kya Hoga Kalia?" to be more apt than "Kitne Aadmi The?"

Dinesh Gopalan said...

Jaishankar, you are right. maybe I should change the title...