Perhaps the way to look at it is that the odds of your first guess being right never improve. They were 33% when you guessed, and they remain 33% when you are offered the choice to switch. The switch gives you odds of 66%.
Exactly! 
This problem comes up often on the net and real life. I think it’s hard for people to understand the true odds because 3 doors is pretty close to 2 doors. There’s not enough difference for it to be obvious it’s worth it to change.
I think it’s easier to understand the problem if you consider a lot more doors than just 3. Imagine there are 100 doors. You initially pick door #10. The host then opens 98 doors to empty rooms and leaves door #52 closed. If the odds were still 50/50, then you would win half the time by keeping your first choice. But that’s not the case. You’ll only win 1/100 times if you keep your same door and you’ll win 99/100 times by switching doors.
With a lot of doors, it’s easier for people to see that their first choice didn’t get any better and it’s much more likely the prize is behind the door that the host didn’t open.
I think the easiest way to understand it is to realize that you are given the choice of one door or two doors. When you name one door you can have what is behind that door, or you can have what is behind both other doors. Since there is just one prize, clearly the prize will be behind one of the two doors twice as often as the one door.
When you name a door to start with you are dividing the doors into two sets. A set with one door, and a set with the two other doors. Monty is giving you the chance to select either set, but first he opens one empty door in the set of two, and then he is asking if you want the third door. When you ‘switch’ all you are doing is saying you want what is behind both doors in the set of two, knowing one of them is empty, and knowing which one it is.
Imagine if Monty started it all by saying you could pick two doors. You pick two. Obviously the prize is behind one of those two doors 66% of the time. Then he opens one of the doors that is empty and asks you if you want the unopened door that you picked, or the one door you didn’t pick in the beginning. Nothing has changed about your odds. The two doors you selected had the prize behind one of them 66% of the time and it still does. Now he’s asking you if you want to switch to the one door that had only a 33% chance of having the prize behind it since the beginning, or the unopened door that was part of the set of two with a 66% chance of having the prize.
Both Filmore and TriPolar frame it nicely in their own respective ways.
Has he said more stupid shit?
Holy shit. I didn’t think I could think any new thoughts about the Monty Hall problem but you just gave me one. Thanks!
Exactly as has been explained much more clearly several times already in the thread. Your own statement is full of ambiguities and other ill-defined terms.
I would go further and say that the post that reopened this thread was flat out wrong, mathematically speaking.
Going back and looking at the post where I brought up Klosterman, I must say I think my explanation of the Monty Hall problem was pretty good, even if I do say so myself: http://boards.straightdope.com/sdmb/showpost.php?p=14775744&postcount=161
Why would you bump a 3-4 year old thread with this nonsensical dreck? The problem has already been very well covered and it’s very simply put: 1/3 chance of getting it right if you stay, 2/3 chance of getting it if you switch. There’s no more to it than that.
A lot of people need a little more than that, because they don’t see *why *that’s true.
How about this more concise version: If you think you picked the right door the first time when there were three to choose from, stay. If you think it’s more likely than not that you picked the wrong door, switch.
Sure, but that’s the part that’s been very well covered already. He just bumped the thread with all this meaningless gibberish that attempts to complicate the matter more than need be.
True enough! And, ironically, claims to “clarify” the matter. :smack:
It was immediately followed by the simplified version - The 2 Final choices appear to be 50%/50%, but at the odds of the FIRST guess (between 3) is still only 1/3 (33%)
Perhaps you didn’t read the rest? It was obvious the first message didn’t break it down into understandable terms, but the concept was basic. The followup should have made that more clear for you.
That’s just a restatement of the solution, and it used terms that weren’t rigorously defined. Choices between two selections always “appear to be 50%/50%” if you don’t understand math. A key point is the odds presented originally (1 in 3) don’t change when MH reveals the goat because no new information is given.
Your simplified version doesn’t help explain the logic of the problem, and it serves to confuse things missing the important points. Everyone responding here understands the probabilities of the original problem. This old and well worn thread was really about explaining to people who didn’t understand and your post really doesn’t advance the discussion in any meaningful way.
That’s precisely the method I used to finally grasp it a couple of years back. I just could not see it before scaling up; it then became crystal clear.
That’s exactly what I said. I was trying find a very simple way to explain this to someone who didn’t already understand the concept. And, why the odds of sticking are only 33% vs 66% and not 50%/50%, as they may appear to be. There were so many discussions, it wasn’t easy to follow all 4pages of detailed examples. Sorry if my post wasn’t good enough for you. Just pass it by. 
4 pages?! How about at least 20 threads going back over 15 years. Enjoy some light reading. 
http://boards.straightdope.com/sdmb/showthread.php?t=704074
http://boards.straightdope.com/sdmb/showthread.php?t=527671
Was there a new breaththrough in mathematical theory?
[Reads posts . . .]
Oh, nm.