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)
- 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]