Last time I started out talking about this, I was sidetracked. I'll try again.
You are on a game show. There are three doors. Behind two of the doors one would find a worthless gag gift. A pile of shaving cream or whatever. Behind the third door, a big check for a million dollars.
You pick door 1, aware that the probability of getting the $1 million by virtue of this choice is 1/3.
The game show host ("Monty Hall" for us oldsters), says, "Ah, that might be right or might be wrong. I'm gonna give you a clue and open door number 2."
Door 2 opens. You see a pile of shaving cream. Everyone laughs because by convention the sight of a useless 'gift' is funny.
Monty renews the choice. Now narrowed down a bit. You can stick with your initial decision, Door 1, or you can go with Door 3.
What do you do and why?
The striking answer is that if you are a rational being you will switch to door number 3 at this point.
Your odds of being right with the initial choice were 1/3. Your odds of being right with that choice are STILL 1/3. The odds that the good prize is behind the only other closed door left, then, are 2/3. So that's the way to bet.
Does this strike you as odd? It strikes everyone as odd. But consider the importance of the fact that Monty Hall must have known what was behind which door in order to make the proper Reveal. His decision was not random, and that makes it a clue for yours.
Tell me how I am missing the point. Once you know that the check is not behind Door 2, it's 50-50 that it is behind Door 1 and 50-50 that it is behind Door 3. Monty Hall's opening of Door 2 revealed nothing about whether Door 1 or Door 3 is the more likely winner.
ReplyDeleteAh, but it did reveal something. He is curating the doors. He must know which door the check is behind. He MUST open door number 2 if the check is behind door 3. He may open either door if the check is behind number 1. So his choice to open door number 2 gives you moderate though of course non-decisive indication that the right answer is 3.
DeleteIt is easy to get confused because there are so few doors. Suppose there were a million doors. (So, to make it worthwhile, the check is for a billion dollars.) You guess Door Number 33,479. Monty says, "your billion might be between 33,479. Or maybe not. As a clue, I will now open 999,998 of these doors." Nothing is behind ANY of the doors he proceeds to open. So the check must now be behind EITHER door 33,479 or, say, 252,391. You'd probably go with 252,391!
A whole book has been written about this, by the way. https://www.amazon.com/The-Monty-Hall-Problem-Contentious/dp/0195367898
I think that I grasp your argument. In the 3-door scenario, the initial pick had 1-in-3 chance. After Monty Hall opened Door 2, Door 3 had 50% chance, making it more likely than Door 1.
ReplyDeleteIn the million-door scenario, the initial pick had a 1-in-a-million chance. After Monty Hall opened 999,998 doors, the one he left closed has a 50% chance, making it MUCH more likely than Door 1.
It still appears that, in both scenarios, after only two door remain, they each have a 50-50 chance. But somehow, that doesn't seem true in the million-door scenario. But I still do not fully grasp it.
You may be interested in the online simulations of the game. Here's one: http://onlinestatbook.com/2/probability/monty_hall_demo.html. It doesn';t use a check and shaving cream -- it uses a sports car and goats. Anyway, I just wasted a little time using the "stay" strategy the first ten times and the "switch" strategy the second ten. I won only 20% of the time using a "stay" strategy and I won 40% of the time switching.
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