I read that question, and back-and-forth answers/comment section, with much amusement…boy did that create a firestorm! lol…
But in reality, it’s soooo much simpler than all that.
When Cecil said that Monty Hall is no dummy, and that empirical evidence would give rise to contestants figuring out what to do based upon Monty’s choosing whether or not to reveal a door (based upon a contestant’s initial choice being right or wrong) is, in and of itself, wrong. Not that Monty Hall isn’t a dummy (haha), but that it doesn’t matter what Monty chose to do, other than playing head games which can’t be factored in to probability analysis. I’ll explain…
When a game of chance, such as Let’s Make a Deal, involves only 3 choices…one of which is going to be revealed as wrong by someone “in the know” (as Monty obviously is)…it’s a special case. When you only have 3 choices, then any one you pick are 33% apt to be right. This means they’re 66% apt to be wrong (oh quit bickering over that last 1% lol).
When Monty reveals, with foreknowledge, a remaining wrong choice after you’ve chosen…he has effectively removed 50% of the 66% potentially right choices (1 of the other 2 choices you didn’t select…and at least one will always be wrong). That ends up being 33% remaining probability of being right, which were **your odds **from the get-go should you switch. In a nutshell, it makes no difference if you switch choices. QED.
Once again, assuming Monty is always, and without fail, going to reveal a knowledgeable wrong choice after you’ve chosen from only 3 alternatives. Now…this doesn’t hold water for any amount of choices above 3, where the “Monty” type person in the know is going to reveal all the remaining wrong choices except 1 after you pick something. For example:
Let’s say Monty gave you 4 choices…you pick one, he reveals 2 wrong choices…leaving only one other choice besides your own…you originally had a 25% chance, but now you have a 37.5% chance by switching because, out of the 75% potential right answers versus your original one, he just removed 66% of them. (2 outta 3 removed, and those 3 comprised 75% probably of being the right choice…and 2 outta 3 is 66%…66% of 75% is 37.5%…clearly you should switch).
In the case of Marilyn vos Savant’s extreme example…then clearly the odds improve astronomically by choosing the remaining choice other than your own. To wit…she brought up 1,000,000 doors, and you choose 1 then she shows you 999,998 wrong ones except for number #777,777. So now it’s a contest between your initial choice, which we’ll call #1…and her remaining option of #777,777. Your initial choice was 1 out of 1,000,000…which is .0001% chance of being right. Not very hot odds, though Vegas would love them.
Her initial odds were 99.9999% of having the right door…and, nice lady that she is…out of her 999,999 doors, she decided to remove 999,998 of them that are guaranteed to be wrong Which means she removed…well, it’s a big percentage…but suffice it to say 99% of them. lol
So out of your initial .0001% chance of being right, you now have the opportunity to decide to take a gamble (haha) on being 99% right (when the math is done over the percentage that she had, then removed, and vis a vis your original percentage). Yeah…not too tricky, there.
So…to conclude. In the case of Let’s Make a Deal, when you choose something and every other wrong option but 1 is left…and the initial choices are only 3…it doesn’t matter. This is, of course, assuming that Let’s Make a Deal mattered in the first place.