I enjoyed reading this and it's certainly well written, but I must admit I disagree with your conclusion.
"Those kids proved that a group of people through several rounds of choices had a collective average probability of 66.666% of picking the right door if they switched doors in a thousand-round system. They did not prove that the INDIVIDUAL increased their chances of winning to 66.666% by switching doors in a two-door-one-choice-ONE-ROUND system."
I guess I just don't see how you're distinguishing between the statistical probability and the in-the-moment probability. I think the confusion might be arising because the problem excludes all circumstances in which the first door opened by the host reveals the car and that we've already lost.
The effect is more obvious with more doors. Imagine its 10 million goat doors and one car door. We choose door #1 and the host opens all but one of the other ten million doors. Obviously, in most cases (99.9999998%) he's going to end up opening a door with the car behind it, revealing that we've lost. The other .0000002% of the time, he opens nothing but goat doors and leaves one door untouched. Your original door and this other door both had a .0000001% of having the car when the game started, but now we've watched ten million doors open and reveal goats, all the while ignoring one random specific door besides your own... History shows us that if we let this scenario play out millions of times, 99,999,999 out of 10,000,000 who choose to switch doors end up being correct, while 99,999,999 out of 10,000,000 who choose to keep their original door end up being wrong-- so it would seem to me wise to go with the 99.9999998% success rate of switching doors, or likewise, the 66% success rate of switching doors in the original formulation of the problem.
Glad you disagree! Feels like the left and right sides of my brain (so to speak) are having a debate, and i can't tell which is which!
i'm pretty sure i follow your logic... Which has an answer to the "mystery" i mentioned: What mechanism causes cars to be placed behind unpicked doors more frequently than picked ones? Your answer, if i understand, would be something like, "The mechanism is that there are more unpicked doors than picked ones. It's simple and straight-forward. Cars are scarce man, there ain't enough to go around. And goats? Well, goats are abundant, but nobody wants a goat do they? It's the system man... Fight the power!" And i have a special place - either in the cockles of my heart, or in the left or right side of my brain, though possibly in my corpus callosum - for that answer... And i did see an explanation of that thousand door variant.
"How do you distinguish between the statistical probability and the in-the-moment probability?" i actually can't tell if the in-the-moment probability is 50%, 66.thedevil%, or indeterminate. What distinguishes them, i think, is that the in-the-moment case, the real case, has no "population" of statistical significance. Neither a population of choices, nor a population of rounds, nor population of people, which are the same thing really. The population in-the-moment is just one. And since it's just one, i think that it is a case of the "Ecological Fallacy", or a "Fallacy of Averages" or "The Statistical Fallacy" if there is such a thing as either of those, to apply probabilities derived from a different population size.
Doesn't some other factor in-the-moment have to determine the probability... Er, is "probability" per se even determinable in a single case with a population of one? Put it this way, if i run the game a dozen times the potential probability that i get a DOZEN correct answers improves if i do NOT follow the rule of always switching doors, since the more times i run the scenario while always switching doors actually destines me to ONLY being correct two-thirds of the time as the round count approaches infinity. Therefore, many of my teachers found me rather annoying back in school.
Now, i might not actually be doing statistics when i reason this way. But are we doing statistics when we say, "Hey Bob, switch doors and your chances improve." if he only has one chance? i'm not sure if all of my reasoning is sound, but i believe my conclusions -1. That there is a fallacy. 2. What the kids actually proved - are roughly correct nevertheless.
But i'm no mathematician! i just don't believe anything; in the sense that if i can't reason something out for myself than what i've got is a hypothesis, not a certainty, and if i can reason it out for myself then i still have a hypothesis, since new data might change things.
i maintain that: If "Bob" gets a goat because he switched doors, we erred in advising him to switch. It's starting to seem like a pedantically trivial point...
(It's occurring to me that an unknown ancestor somewhere in my family tree may have been a troll... The horror! The shame!)
...But i think it has something to do with the creative power and existence of an individual, which any "authority of concepts" can't touch, whether mathematical or otherwise. i think it's because concepts aren't true, they're just useful for some purposes sometimes.