Movie "21" MIT classroom probability problem....one of the best problem of probability i came across

Replies

• 

  Shashank Moghe

  Continuing our previous discussion, I still feel the probability of getting the car after the first door opened as goat is
  
  (2/3)[probability of getting a goat] *(1/2)[probability of getting a car] = 1/3
  So, probability of getting a goat and a car is 1/3. Surprisingly, if I want 2 goats, the probability is still 1/3.
  
  Well, something is definitely wrong here. What do you think?
• 

  Shashank Moghe

  Well, this got me reading, and a non-mathematical proof is given on Wikipedia:
  
  Solutions[#-Link-Snipped-#]
  Simple solutions[#-Link-Snipped-#]
  The solution presented by Monty Hall Problem Citerefvos Savant1990B) in Parade shows the three possible arrangements of one car and two goats behind three doors and the result of staying or switching after initially picking door 1 in each case:
  behind door 1behind door 2behind door 3result if staying at door #1result if switching to the door offered
  CarGoatGoatCarGoat
  GoatCarGoatGoatCar
  GoatGoatCarGoatCar
  A player who stays with the initial choice wins in only one out of three of these equally likely possibilities, while a player who switches wins in two out of three.
  
  Source: Wikipedia (search "Monty Hall problem")
  
  
  I want someone to mathematically back this result on CE.
• 

  Garvit Garg

  i have mathematical solution but still want you to work on it....i dont want to ruin question by putting out solution this early....
• 

  Anoop Kumar

  He given the answer with explanation.. and it's right to switch the gates. What is the discussion here?
• 

  Garvit Garg

  how the probability is 66.67% if we switch gate???
• 

  Garvit Garg

  Shashank Moghe

  (2/3)[probability of getting a goat] *(1/2)[probability of getting a car] = 1/3
  Well, something is definitely wrong here. What do you think?

  yea there is something wrong here....think over it....
  hint: think on (1/2)[probability of getting car]
• 

  Anoop Kumar

  Garvit Garg

  how the probability is 66.67% if we switch gate???

  Are you saying probability on switching gate would be 1/2.
  
  Suppose if you have 4 gate and you are allowed to choose 3 answers, what would the probability of getting car 3/4 isn't it!
  
  Now, if another way of answering like double-dip, with each answer I will tell you that you that car is there or not. In this case , Are you saying after choosing the 3rd time, overall probability would be 1/3 because there is only two gates left and car can be in either one.
  Then, there always probability of 1/2 because either you loose or win.
  Can't think of simpler explanation.
• 

  Garvit Garg

  i m not able to understand what u want to say....let me make myself clear...i think probability ll be 66.67 % if we switch gate....if u agree than explain how if u dont than explain how....that the question....
• 

  Anoop Kumar

  Ohh.. I thought you are saying answer is 50%.
  But again that guy explained in simple way.
  Ok here,
  Edit: Misleading explaination, removed it.
  First, probability is 1/3. now anchor open the gate and showed that car is not there. Now if you stick with your answer probability is still 1/3 but if you change the answer, you have selected two answer for the question and that is why probability becomes 2/3.
  
  Now here is mathematics:
  first answer : 1/3
  second answer after opening gate: 1/2 * 2/3 = 1/3
  Total probability: 1/3 + 1/3 = 2/3
• 

  Garvit Garg

  Anoop Kumar

  First, probability is 1/3. now anchor open the gate and showed that car is not there. Now if you stick with your answer probability is still 1/3 but if you change the answer, you have selected two answer for the question and that is why probability becomes 2/3.

  Ben never get to select two options....he gets to switch his option....so at the end he is not with two options selected....maybe u need to know the case nicely...
• 

  Anoop Kumar

  Hands down captain 😁,
  For me that solution in video is simplest one. I have seen problem and explanation well enough on NatGeo first before watching this movie.
  Here is another explanation if there are 1000 doors
  

  What if there were 1,000 doors? You would have a 1/1,000 chance of picking the correct door. If Monty opens 998 doors, all of them with goats behind them, the door that you chose first will still have a 1/1,000 chance of being the one that conceals the car, but the other remaining door will have a 999/1,000 probability of being the door that is concealing the car. Here switching sounds like a pretty good idea.

  Source: #-Link-Snipped-#
• 

  Shashank Moghe

  Garvit Garg

  i have mathematical solution but still want you to work on it....i dont want to ruin question by putting out solution this early....

  
  If you are talking about the mathematical solution mentioned on the Wiki page, that was way over my understanding of conventional symbols. I do not want to subscribe to the same assumptions about the "role of the host" in these calculations. That is where the paradox lies.
• 

  Shashank Moghe

  Anoop Kumar

  Ohh.. I thought you are saying answer is 50%.
  But again that guy explained in simple way.
  Ok here,
  First, probability is 1/3. now anchor open the gate and showed that car is not there. Now if you stick with your answer probability is still 1/3 but if you change the answer, you have selected two answer for the question and that is why probability becomes 2/3.
  
  Now here is mathematics:
  first answer : 1/3
  second answer after opening gate: 1/2 * 2/3 = 1/3
  Total probability: 1/3 + 1/3 = 2/3

  In the second chance both the doors are equally mysterious. How can you raise the probability of changing to 2/3?
• 

  Anoop Kumar

  Anoop Kumar

  Source: #-Link-Snipped-#

  I tried to explain from my side as simple as it can be. Also given the Mathforum page , I think they are more intelligent in Math (At least better than me😉).
  I can't bring Ramanujam here ☕
• 

  Shashank Moghe

  Well, that is not necessarily true. Everybody posting on Mathforum doesn't need to be a mathematician. Even more so, a mathematicians word isnt always true. You can be right and he/she can be wrong. That is the beauty. As for the link, I am reading it now.
  
  As for your explanation, at the very least you were not able to reproduce what the Mathforum postor wanted to say.
• 

  Anoop Kumar

  I agree my above math explanation is misleading. It's more about paradox.
  Ok here is the game itself, explanation by Monty hall and controversy and real results, play the game 1000 times and see the result.
  #-Link-Snipped-#
  
  My view, this problem is like Schrodinger's cat, brain is conflicting with maths proof.
• 

  Shashank Moghe

  Let us not go there. This is in no way like Schrodinger’s cat. There, there is no conflict between the observation of the brain and mathematical proof. Here, the role of the game controller is brought into the mathematical proof, I find that disconnected.
• 

  Garvit Garg

  let me try explaining solution this way....
  we are counting probability of getting right answer if we switch gate......
  so now the case is....we choose a gate, than, anchor opens one door out of remaining two(which is not having car, obviously) than we will switch the gate ....
  now if we choose right answer when anchor asks us to choose gate out of three gates than anchor ll open the one of the wrong gate(out of both the wrong gates) and we will switch the door to another wrong gate....and we will lose the car.....
  other than this condition....if we choose a wrong gate out of two wrong gates than anchor ll open another wrong gate and we will switch our answer to the right gate....
  so basically if we choose wrong gate in first attempt than we will win a car for sure....and if we choose right gate in first choice than it ll lead us to wrong gate....
  and probability of choosing wrong gate in first choice is 2/3, giving us car at the end.....🎉
• 

  Shashank Moghe

  Garvit Garg

  let me try explaining solution this way....
  we are counting probability of getting right answer if we switch gate......
  so now the case is....we choose a gate, than, anchor opens one door out of remaining two(which is not having car, obviously) than we will switch the gate ....
  now if we choose right answer when anchor asks us to choose gate out of three gates than anchor ll open the one of the wrong gate(out of both the wrong gates) and we will switch the door to another wrong gate....and we will lose the car.....
  other than this condition....if we choose a wrong gate out of two wrong gates than anchor ll open another wrong gate and we will switch our answer to the right gate....
  so basically if we choose wrong gate in first attempt than we will win a car for sure....and if we choose right gate in first choice than it ll lead us to wrong gate....
  and probability of choosing wrong gate in first choice is 2/3, giving us car at the end.....🎉

  
  This is the verbal explanation. Statistically, it is very beautifully (and simplistically) tabulated on the Wikipedia page. I am talking about the mathematical proof.