re: Marilyn vos Savant and the history of the Montel Hall question

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In the linked article, they explain that
"The short answer is that your initial odds of winning with door #1 (?) don’t change simply because the host reveals a goat behind door #3; instead, Hall’s action increases the odds to ? that you’ll win by switching."

To me, they are admitting that the "initial" 1/3 odds do not change simply by dropping the goat. I read it has saying that the Host allows you an extra chance of winning by giving you another bite at the apple when you consider that the Host will never drop a car so you effectively get a 2nd chance to pick the car, with 1 less goat to choose from.

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This is correct.

We know a goat is behind one of the doors we didn't choose. The presentation of the goat is actually a trick because it shows us what we already knew, but causes us to change the odds.

50/50 at that point.

Everyone is making the flaw of focusing on what you pick where, logically, it makes more sense if you focus on what you don't pick.

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Again, the problem with this riddle is that the mathematical explanations always start with the premise that you must carry forward the old 33/66 probabilities from the first problem into the second. That is the cheat--you don't. The Monty Hall problem really is a 33/66 probability problem changed into a 50/50 problem, but discussed mathematically (incorrectly)after the basic premise has been changed as if it is still a 33/66 problem.

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Your logic here is incorrect, and the reason so many people have trouble with it.

It is still a 33/66 problem after one goat door is eliminated.

Look at it this way.

Host: You can either pick one door or two doors.

You obviously pick two doors.

Host: But wait; if you pick two doors, I will show you a goat behind one of your doors before opening your other door. Would you like to change to just one door?

Since you already know that at least one of the two has a goat, you wouldn't care and would go with the two doors anyway.

That's the exact math behind the problem as is. You are choosing between 1 door or two doors, one of which is revealed to contain a goat.

The odds improve to switch doors only because Monty always opened a door that didn't have the winning prize behind it first, in order to build up the drama.

If Monty had always just randomly selected a door to open first then sometimes the first door opened by Monty would contain the prize and the excitement would then be over before the contestant had his choice opened. They never did it that way, thus Marilyn knew that the door that the contestant didn't pick and the door that Monty didn't pick had a 2/3 chance of having the grand prize behind it.

The only way the contestant could lose by switching is if he choose the correct door to begin with when he had a 1/3 chance of being right. So switching choices after Monty had eliminated one of the other doors to have the goat gives the contestant 2/3 chance to win instead of sticking with his original 1/3 chance to win.

I'm floored that math professors couldn't figure that out. (of course most were from Florida, so that makes it more understandable)

Regardless of how far out you want to extrapolate it (1000 doors, etc.), if you NEVER change your pick, Monte will always eliminate everything EXCEPT 1 car and one goat, leaving you with a 50/50 shot.
Likewise, if you change your pick EVERY time Monte exposes a new goat, he's still going to leave you with a 50/50 shot in the end.

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if 1/3 of the choices are randomly taken away

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This is not correct. Monty does not randomly choose, he only picks a door with a goat.

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The odds improve to switch doors only because Monty always opened a door that didn't have the winning prize behind it first, in order to build up the drama.

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Yeah. It would have been far less exciting if he chose the car before there was a chance to make the decision.

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Yes, which is why I clarified my agreement "conceptually" versus reality.

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Conceptually and realistically (they've done extensive simulation studies) the odds are 66/33.

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If you picked the car correctly (your 1/3 pick), you will now be given yet another chance to screw up and change to the wrong pick but this time you have a 50/50 chance instead of the "initial" 1/3 chance.

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There is never a 50/50 chance though. Just because there are two options, doesn't mean their odds are equal. If you played this game enough times to get a large sample, you would win 2/3 of the time by switching.

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Conceptually and realistically (they've done extensive simulation studies) the odds are 66/33.

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Hence my observation:

"With all that said, the explanation of the solution is provided in the article and there is nothing, that I know of, that any of us can do to refute the results."

The actual results are what they are. There is nothing to discuss, except conceptually with regard to one's view of the presentation of the choices made available to the contestant.

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Let me try explaining it this way.

Pretend you are the contestant and you make your pick first between the three doors.

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pretty good scenario, but what about this one...

imagine you grew up the 2nd of 3 brothers in a small suburb. you spent your summers riding your bike to the park and stealing chrome caps off car tires. your oldest brother was many years apart from you and went away to college or died or something. you didn't really know him anyway, and he was a dick to you growing up, so it doesn't really matter. but your younger brother was born with a debilitating mental handicap, and you always saw yourself as his guardian and caretaker. it was tough growing up with him because his affliction could be a burden on you. it was particularly embarrassing when mom would drop y'all off at the mall, and you'd try to talk to some girls in spencer's and they'd get all creeped out by your brother smiling at everything and doing weird shite and they'd be like "what's his problem? is he retarded or something?" and you'd be like "shut up, it's not his fault. come on jason, let's get outta here..." his name was jason btw. so fast forward 40 or 50 years, and both of your parents haven passed away now. you're married and you have kids of your own to take care of. not to mention you only have a modest 3 bedroom house, and 2 of the 3 are already sharing a room. someone needs to take care of jason, but honestly you don't have the money or the time. you decide to take what little inheritance you received from mom and dad to the casino and try to win big and put jason up in a nice home close to you so you can visit him a lot. there's a new game at the casino where they have 3...no, a million cards flipped upside down. one of them is a million dollar grand prize. you can pick any card, then 999,998 incorrect cards are flipped over. you're given the choice to stay with your original 1-in-a-million chance or switch. remember, you're all jason's got, and he's depending on you.

so do you see that you should switch now?