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Topic: Monty hall Spinoff(IIm interview) (Read 1372 times) |
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EagleKing
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Monty hall Spinoff(IIm interview)
« on: Jul 16th, 2008, 11:41pm » |
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Let us consider a spin-off to the Monty Hall problem:
one with two contestants and four doors and one host(M.H)
Let the players be called P1 and P2.
So in the final round the 2 players and MH are in front of 4 doors.One contains a car and the other 3 goats.
Player 1 has an advantage of winning the car (bcoz of higher score in the last rounds)such that he is given to choose a door first and tell it to the Whole Audience(including P2 and M.H).However P2 has to make a choice such that he will choose a door other than the one of P1,and he will tell only M.H of it(P1 won't know).
So M.H(who knows whats behind the doors) opens two doors to reveal goats.Of the Two unclosed doors remaining one is definitely of P1.The other one may or may not be of P2 because M.H randomly chooses whether to open or close P2's door.
Now lets consider a situation such that the other door IS THE ONE OF P2.Now P1 is given a choice to shift.
Should P1 “shift” for a better probability?(3 times the other probability?)
If yes then is it true that logically given a choice P2 should shift as well for more probability ?
Or should he stick to his own choice?
Do they shift between the same doors for more probability?
What are the respective preferred choices for P1 and P2?
this question is supposedly an interview question for Indian
Institute of Management(i am not too sure about it's origins)
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towr
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Re: Monty hall Spinoff(IIm interview)
« Reply #1 on: Jul 17th, 2008, 12:13am » |
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on Jul 16th, 2008, 11:41pm, EagleKing wrote:| Should P1 “shift” for a better probability?(3 times the other probability?) |
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yes
Quote:| If yes then is it true that logically given a choice P2 should shift as well for more probability ? |
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No, because there was a chance his door would have been opened. P1's first door remains at 1/4 probability because its situation didn't change, it was excluded from the process that shifted the joint 3/4 probability of the other three doors onto just one of them.
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| « Last Edit: Jul 17th, 2008, 12:17am by towr » |
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towr
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Re: Monty hall Spinoff(IIm interview)
« Reply #2 on: Jul 17th, 2008, 3:41am » |
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(via PM) Quote:U Wrote:
"No, because there was a chance his door would have been opened. P1's first door remains at 1/4 probability because its situation didn't change, it was excluded from the process that shifted the joint 3/4 probability of the other three doors onto just one of them."
(//Also see that we r considering only the case where only the doors chosen by P1 and P2 remain.)
This is considering the problem from P1 's point of view
However if we see P2's point of view:
Probability of first chosen door=1/4
Probability of other door =3/4
Should not he shift??
Plz tell me where is the fault in the above argument(I AM SURE THERE IS ONE).Plz help.......  |
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Perhaps the easiest way to consider this problem is to write it out.
Without loss of generalization, we can say that initially P1 chose door 1, and P2 chooses door 2.
We can have either:
1) car goat goat goat
2) goat car goat goat
3) goat goat car goat
4) goat goat goat car
In the first case, there are 3 ways to leave one of the last 3 doors close; in the other three cases there is only one way (because the car must be left)
So we get
1.1) car goat goat goat, probability=1/12 (=1/3*1/4)
1.2) car goat goat goat, p =1/12
1.3) car goat goat goat, p =1/12
2.1) goat car goat goat, p =1/4
3.1) goat goat car goat, p =1/4
4.1) goat goat goat car, p =1/4
Given that P2's door wasn't opened, we know we are dealing with cases 1.3 and 2.1; and 2.1 is three times as probable as 1.3, regardless of "point of view".
In fact we can ignore P2's first choice, and just give him 1/3 probability that he was allowed in on the game after Monty opened two doors (his choice and Monty's are independent). Clearly he should make the same choice as P1 from that point. (Having the exact same information available)
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| « Last Edit: Jul 17th, 2008, 3:43am by towr » |
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