Friday, December 04, 2015
Monty Hall problem keep popping back up recently, most of them with a catchy headline “even geniuses/PhD got it wrong”.
Some people made their names by sounding as if she is smarter than the smartest Mathematicians by claiming "they were all wrong" (i.e. here: The Time Everyone “Corrected” the World's Smartest Woman and here http://marilynvossavant.com/game-show-problem/)
Is she really "Smarter than the smartest". I kind of call it an intellectual scam. Here is why:
(according to Wikipedia)
"The behavior of the host is key to the 2/3 solution. Ambiguities in the "Parade" version do not explicitly define the protocol of the host. However, Marilyn vos Savant's solution (vos Savant 1990a) printed alongside Whitaker's question implies "
So she implied something in her solution that was not explicitly mentioned.
Ok, according to Wikipedia, she later corrected (clarified) her question in a later post (which made her look the correct one), however the damage is already done, and most people argued against her "implied" on her original posted question which is still available to see on her website:
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?Craig F. Whitaker
For this particular question text, take it literally without assuming informations out of the text, the answer should be "I don't know" like most Mathematicians PhD argued. (see host behavior section of the Wikipedia page) But
Marilyn's solution has its own merit.
What does that mean? Why most Mathematicians got it "wrong"?
Well, there is a thing called Mathematical model ... basically that's the process of expressing a question in mathematical equations. Here is what went wrong. Marilyn built a mathematical model that "implied" the player KNOWS the host will always open a door with Goat. The other PhDs, did not build this into their model.
That's why Paul Erdös kept asking a "Common Sense" solution -- he was able to figure out it, but unfortunately for whatever reason Andrew Vazsonyi refused to understand it, instead he decided to stick with his decision tree approach which means he stuck with his mathematical model refusing to understand if his model correctly reflects the actual question. (https://web.archive.org/web/20140413131827/http://www.decisionsciences.org/DecisionLine/Vol30/30_1/vazs30_1.pdf)
So those "smarter than smartest" people have been implying information that was not explicitly provided in the original text, and they even built computer simulations with this assumption built in!
Since Paul Erdös' commonsense solution was lost by Andrew Vazsonyi, here is my common sense explanation and why I call Marilyn's posts intellectual scam.
Here is why "Smart people" get it "wrong"(different understanding than Marilyn):
There are 3 doors, one of them has a car behind it. you didn't know which one, so you randomly chose one, you have 1/3 chance winning the car (no dispute here!). Now, the host will open a door, there would be the following possibilities:
1) you selected car; host opens a door to (either one is) a goat - 1/3 chance
2) you selected 1 of the 2 goats, host open the other (only) goat and leave the car closed - 1/3 chance
3) you selected 1 of the 2 goats, host open the car and prove you got a goat and leave the car closed - 1/3 chance
OK, the interesting thing comes here:
According to this text, the host opens another door shows you a goat -- was that by chance? IF he opened the goat by chance, which means you were lucky we are not in scenario 3) above, then the 2 left over options are 50:50 -- the PhDs are correct, switch (get 2,) or not (get 1,) is no difference.
Then why Marilyn could prove she was correct? Well, she proved her solution (switch) is best solution for the math model she built HOWEVER, it is NOT exact reflection of the actual text. (here in the explanation of her solution she added "...and the host always opens a loser. ") This is the key of her "scam", this "always" was not given in the original question (mathematically this means she is adding additional conditions and restrictions and changed the actual model).
What she added "the host always opens a loser" basically eliminated possibility 3) above!!! (Right! it means between 2) and 3), the host will always use his knowledge to eliminate 3) and leave you with scenario 2) only. IF you KNOW he is forced by rule to do this(help you eliminate 3), then switch is the best strategy because in 2/3 chance where you make wrong choice in first place, the host is forced to take half of the possibility (scenario 3) away before he ask you if you want to switch. So Marilyn is correct now to solve her mathematical model and she can prove it, even with computer simulations. However, this rule was not given in the original question text.
So now here is the commonsense description of a properly construct the question that Marilyn solved:
a) a car is randomly placed behind 1 of 3 doors, the other two hide a goat each;
b) you have zero information to begin with other than a);
c) you can chose one of the 3 doors as round 1;
d) the host will then have to open a door with goat (he is forced by rule to do so, and he is not allowed to open a door with car behind it, he cannot chose to skip opening a door either, and you KNOW this rule), this is round 2;
e) you are then asked "switch or stay", this is round 3.
And here is the Commonsense Solution:
1) for round 1 above, do you agree you have 1/3 chance winning, 2/3 losing? (of course, no one denies that, right?)
2) now it comes the interesting part, because of "host always open a loser", IF your first selection was a loser, then the host is FORCED to open the other loser (so by now both you and the host are holding on to a loser door, but there are only 2 losers, so the leftover must be the winner) And it does not matter which loser you first selected, because by the rule, the host will have to eliminate the other loser anyways, so either one of the 2 loser you pick, the host had to remove the other loser and leave you with the winner.
3) so now if I ask you "how likely was your first round choice correct?" the answer of course is 1/3, "how likely your first round choice was wrong?" the answer of course is 2/3.
4) because of the host is forced by rule to eliminate one loser for you, if you were wrong to begin with, the only door left would be the winner. So you should always select "switch" which means "I was wrong first round".
You ask why? I still don't get it? like mentioned before, the host was forced by rule to eliminate one of the 3 possibilities (the 1/3 chance where he opens the door showing the car), and if you know this rule, then here is the "possibilities" (lets say you select door #1)
1) car is behind door #1, host can freely open either #2 or #3, he is not helping you - you have 1/3 chance
2) car is behind door #2, host is FORCED by rule to open #3, so if you switch, you get the #2, winner - you have 1/3 chance;
3) car is behind door #3, host is FORCED by rule to open #2, so if you switch, you get #3, winner - you have 1/3 chance;
See, if you stay, your only hope is it turns out to be 1), 1/3 chance; however, if it was either 2) or 3) above, the host is forced to point you to the winner by open the other loser. So you have two times of these 1/3 chance winning if you switch.
This ONLY works like Marilyn solved IF the host is forced by rule to always open a loser, and you KNOW this rule. Her mathematical model assumed this very specifically, so was her computer simulations -- because the computer model was programmed to "always open a loser", the simulation result supported her claim.
Why it's a scam? Because the original question did NOT specify this "host has to open a door with goat" which in essence change the round 3 to a different question, and most Mathematicians and PhD actually used common sense to think about it like this (without assuming the player knows a rule forces host to always open a goat): If you do NOT KNOW the host opened goat door was the rule of the game (instead, you think you were lucky that he did not open a door with car prove you were wrong at round 2), then there is no reason to switch in round 3"
There have been comprehensive discussion of host behavior and why he opens a door with goat i.e. some variation says host only open a door to goat if he knew you were correct, otherwise he would skip round 2, and ask you if you want to switch without opening a door in which case switch is a sure losing strategy. Of course Marilyn's math model was not built this way.