Amber G. wrote:
Vina et all -
The problems is fairly well discussed, at:http://en.wikipedia.org/wiki/Monty_Hall_problem
As I said before, Your initial chances of success was (1/3) opening of the door by the host gives absolutely NO other information and the probability so it still remains same
(1/3) that it was in the original door. Hence the other remaining door must have prob = 1-(1/3) = 2/3
Sorry Amber G. Your explanation is wrong. Please read the link you posted carefully ( I had no idea about the background of this Monty Hall thing). To say that the host gives NO other information is wrong
. He actually gives tremendous amount of information , over stages. Initially he tells you that there are 2 goats and 1 car and 3 doors. When at a later stage, you touch a door and he opens another door and shows a goat, what he tells you is that the car is either in either the door you are touching or another one, he eliminates
one possibility. That is tremendous value and information. That is exactly what lets you make that switch. If he didn't make open the door and show the goat, you wouldn't switch, because you have NO further information.
The 2/3 and 1/3 that you are referring to and the explanation that the link gives are overall probabilities, which of course don't change . But the CONDITIONAL probabilities are different , when better information becomes available and just like I calculated are 1/6 and 1/3 . They too have shown the decision tree with the conditional probabilities and and shown it on the page in the link you posted.
Dileep, you are right, this problem is all about "timing"/"stages" and remembering across "stages" . If you lose that memory, you end up choosing between the 2 doors after the goat is shown, with equal probabilities of 1/2 and hence the wrong answer of " don't switch ".
The entire thing is quite intuitive. The Wikipedia link despite all the pictures does a bad job of explaining it. Let me take a stab at it in simple english.
At the beginning of the game , there are 2 goats and 1 car and 3 doors. So the chance if you touch a door, what lies behind it is a goat is twice the probability that what is behind it is a car.. ie. 2/3 vs 1/3 .
So you touch a door and don't open it. Now the friendly goat dealer /car dealer opens one of the other two doors and shows you a goat (gives you info) !. Now if you have no knowledge of the starting conditions, you would say.. oh well, 2 doors, 1 goat , 1 car. so equal probability , so let me stick with my choice.
But if you as, wait a minute.. Either the unopened door I am touching has a goat or the other unopened door has a goat . Now I now that initially there were twice as many goats as cars . So when I initially touched the door, the possibility that it was a goat is TWICE that of that of a car. So oh my god, the kind dealer has exposed one more goat. Now if like a schmuck, I open the door I am touching, it is twice as probable, that i will get a goat and not the car. Since you want the car, you switch and open the other door , raising your chances of getting a car.
See , you could switch only because you remembered the intial state of the problem. If you didn't know that there were twice as many goats as cars when you touched , you couldnt make the decision on whether to open or switch.
This kind of thing /decision trees is used a heck of a lot in finance/strat planning . That is the reason I still can do this math with any amount of clarity. There is tremendous value in information. For eg, in a venture investing situation,you would structure the deal in stages ( round 1, round 2 round 3 etc) , with specific metrics in each stage, because you can use better inforamation as time goes by and information becomes avialable to increase your expected pay offs.
Same thing with M&A transactions. You structure many deals with earn outs, based on certain conditions because of availabilty of information over time and as a risk shifting measure.
In project planning /investing in new projects etc, you use Real Options valuations and do this kind of thing to see whether investing makes sense and to get the full value of the project. In fact, if you are not careful and dont value the real options you end up leaving tremendous amount of money on the table or end up making skewed decisison.
Ok, if all that sounds esoteric, let me give you an example we all know.
If in 1989 when the LCA project was launched, say, the DRDO gave out a timeline of 10 years and said, the investment is $10b and the expected payoff is $12 b ,you probably would not put the money right away. You would tell them, okay prove that the stuff works by building a TD program in 3 years and take $2b now. Lets make a go no go decision in 3 years ,depending on what progress gets made and how the TD shapes up. Now if you do that staging, the expected value is probably going to be much higher than the $12b you would have got if you plonked the $10b right away , because, you gain options to kill the project after 3 years if it is lemon with minimal losses, and also the option to invest in it fully and make it a success if it comes good after 3 years. Deferring the $7b investment to 3 years later also has tremendous value (time value of money).