This is an adaptation of the Monty Hall problem into a form that hopefully will be easier for some people to understand. It is not the best way to understand the Monty Hall problem. But if all else fails, maybe this will help.
Even if someone doesn't find this argument convincing, hopefully you can find out which step leaves them unconvinced, and that gives you a more specific problem to try to explain. (When I first wrote this I expected Version G might be the sticking point, which is why I tried to explain it multiple times.)
A few years before Let's Make A Deal, a gameshow host named Lymon Halt added a segment in his show Steal A Mad Leek where he gives a contestant a chance to win $1000 dollars.
Before the show, Lymon got 100 opaque envelopes and numbered them from 1-100. He randomly chose one envelope and placed a $1000 bill in it. The other envelopes were entirely empty.
In the show, he presented the envelopes to the contestant. The constestant could see the envelopes but not handle them and had no way of knowing which envelope contained the bill.
Lymon asked the contestant to choose 50 envelopes. They could be all the even-numbered ones, all the odd-numbered ones, all the ones numbered 50 or less, all the ones between 37 and 86 inclusive, or a list of 50 distinct ones.
Lymon moved the selected envelopes into a pile, the "selected pile", and moved the remaining envelopes into a second pile, the "remaining pile".
Now Lymon gave the contestant a second, final choice: they could have the contents of the envelopes in the "selected pile", or they could have the contests of the envelopes in the "remaining pile".
Since the contestant didn't know which envelope contained the $1000 bill, and assuming Lymon chose randomly when he put the bill in, there was a 50/50 chance the bill was in the "selected pile". Therefore, there were equal odds the bill was in the selected pile or the remaining pile: the final choice had a 50/50 chance of getting the bill.
Once the contestant chose, Lymon made a big show of going through the chosen pile, opening the envelopes one by one, delaying the discovery of whether the contestant had chosen the right pile. Since Lymon knew which envelope the bill was in, if it was in the pile the contestant chose, Lymon could avoid opening it immediately to prolong the suspense.
The producers of the show complained that this segment was taking too long, especially the part where the contestant listed out 50 unique numbers to create the selected pile. Could he skip that part?
In the new version, Lymon had the contestant choose only one envelope. That envelope went into the "selected pile", and the other envelopes went into the remaining pile.
Again, the contestant got a second choice: either choose the selected pile (which had only one envelope), or choose the remaining pile (which had 99 envelopes).
Since the contestant didn't know which envelope contained the $1000 bill, the contestant had only a 1-in-100 chance of selecting the correct envelope. Therefore, there was a 1-in-100 chance the selected pile contained the $1000 bill, and a 99-in-100 chance the remaining pile contained the bill. If the contestant's final choice was the selected pile, they had a 1-in-100 chance of winning $1000; if they chose the remaining pile, they had a 99-in-100 chance of winning $1000.
Of course, if the contestant had gotten lucky and chosen the correct number, then there was a 100% chance the bill was in the selected pile and a 0% chance that it was in the remainder pile. But this doesn't make the chance 50-50; the contestant only had a 1-in-100 chance of getting lucky in the first place, which is exactly why the odds were 1-in-100 for the selected pile and 99-in-100 for the remainder pile.
The producers of the show complained that the segment was still taking far too long. Most contestants chose the "remaining" pile, and because Lymon liked to be dramatic and hold off on opening the one with the bill, that meant he would usually open 90+ envelopes before he opened the one with the $1000 bill. They demanded he skip that part.
This version was the same as the previous up until the envelopes are opened. (The contestant chose a number, that one was placed in the selected pile, and then the contestant chose between the selected pile and the remaining pile.)
If the contestant chose the selected pile, Lymon opened the one envelope and showed the outcome. If the contestant chose the remaining pile, Lymon went straight to the envelope he knew has the bill and opened it.
The odds were unchanged; Lymon was just changing how he presented the outcome. The contestant still had a 99-in-100 chance if they chose the remaining pile.
One contestant of Version C got very unlucky—they actually originally selected the envelope with the bill, but not knowing they had done so, they then chose the remaining pile. Lymon proceeded to go through all 99 envelopes and show they were all empty and the contestant had won nothing. Then, just so people would know the show wasn't cheating, he opened the selected envelope and showed the $1000 bill—then pocketed it to keep for the next show.
The shows' producers were furious that he'd spent all that time opening 99 envelopes that contained nothing. They demanded he find a way to skip that part if it ever came up again.
Lyman, legendary showman that he was, came up with a new way to present the outcome that would handle all of the cases more gracefully.
The setup was the same as before (contestant chooses a number, that envelope is the selected pile, the contestant makes final choice between selected and remaining pile), but he also brought a portable incinerator on stage.
If a contestant chose the selected pile, then Lymon opened that envelope. If it was empty, Lymon also opened the envelope from the remaining pile that contained the $1000 bill and pocketed it. All of the remaining envelopes went in the incinerator.
If a contestant chose the remaining pile and the bill was in it somewhere, Lymon opened the envelope containing the bill, then gathered together the remaining 98 envelopes and chucked them in the incinerator.
If a contestant chose the remaining pile and the bill wasn't in it anywhere, Lymon opened a random envelope--it didn't matter which--then gathered together the remaining 98 envelopes and chucked them in the incinerator.
None of the above presentation changed anything before the contestant's final choice, and as the contestant was still making the exact same choices as before, they still had a 99-in-100 chance of winning if they chose the remainder pile.
A viewer wrote to Lymon complaining that Version D wasn't completely symmetric. If the contestant chose the remainder pile, Lymon would burn 98 envelopes, but if the contestant chose the selected pile, Lymon would sometimes burn 98 envelopes but sometimes burn 99 envelopes (because if the bill wasn't in the remainder pile, Lymon burned all 99 envelopes). Sometimes burning 98 envelopes and sometimes burning 99 envelopes was weird, and the viewer had a simple suggestion for how to fix it.
The viewer proposed that regardless of what the contestant chose, Lymon should always open one envelope from the selected pile and one envelope from the remainder pile. If the $1000 bill was in the remainder pile, he'd open that envelope, but if it was in the selected pile, he should open a random one. Then two envelopes would be open, and Lyman could always incinerate 98 envelopes.
This didn't seem particularly important to Lyman at the time, but he figured he could streamline his presentation a little bit if he did it, so he incorporated the viewer's suggestion.
Again, everything was the same up until the presenting the results (contestant chose one envelope, the chosen envelope went into the selected pile, and the contestant made a final choice), which means the odds were still the same (1-in-100 vs 99-in-100).
Once the contestant chose, Lymon opened up the one envelope in the selected pile and one envelope from the remainder pile, and burned the other 98 envelopes. Of course he chose the right envelope from the remainder pile (if there was one) so the bill wouldn't get burned.
After he had done the above for a week, Lymon made a small change to the presentation. The sequence of events had been:
But Lymon liked there to be as little time between when the contestant won and the end of the segment, so he reversed the order of steps 4 & 5.
Again, the odds didn't change at all, because this was simply a change to the presentation of the results. It's still 99-in-100 to choose the remainder pile.
This also turned out to have extra drama; this way, the bill hadn't been revealed yet when he burned the 98 envelopes, and the audience would wonder if he'd made a mistake and accidentally burned it. (Fortunately, he never did.)
Things had been running this way for a year when Lymon, consummate showman that he was, realized that he could make a small change that might make things more dramatic even though it wouldn't actually change the odds.
The official rule was that the contestant made a final choice between the two piles and got everything that was in the envelopes in the chosen pile. But everyone also knew that at least 98 of the envelopes were empty. After the contestant made their final choice, the very first thing Lymon did was gather up those 98 envelopes and burn them. It didn't matter if the contestant chose the selected pile or the remainder pile; Lymon already knew exactly which 98 envelopes he was going to burn.
So he changed the order of events and burned those 98 envelopes before the contestant made their final choice:
Lymon knew that this didn't change the odds at all, because those were the exact same envelopes he was going to burn anyway. He hadn't given the contestant any new information which which to make the final choice compared to the previous version, because the contestant was going to see Lymon burn 98 envelopes in the previous version too. The only thing the contestant knew now was which 98 envelopes Lymon was going to burn (and which one he wasn't going to burn), but this didn't change the odds that the remainder pile had $1000 in it.
Either the money was in the remainder pile or it wasn't before Lymon burned the envelopes. So if the odds were 99-in-100 the money was in the remainder pile before Lymon burned the envelopes, the odds were still 99-in-100 after Lymon burned the envelopes. Nothing had really changed.
But what contestants saw was now very different. Now, instead of choosing between a pile with a single envelope and a pile with 99 envelopes, the contestant was choosing between a pile with a single envelope and a pile with one envelope unburnt and 98 envelopes incinerated (the contents of which the contestant technically had the rights to claim under the official rules, but since they were empty that didn't mean anything).
Some contestants found it difficult not to see this final choice as 50-50, that this change in the order of events didn't matter. They just saw a choice between two envelopes. But really, they were still choosing between two piles, one with one envelope and one with 99 envelopes. They just being told in advance that 98 envelopes in the second pile were empty--but since there was only one $1000 bill, that's something that everyone who'd played any version of the game from Version B to Version F had known too. They now knew which 98 envelopes were empty, but that didn't affect the odds for whether the bill was in the remainder pile.
Lymon's game show got new producers, and these producers didn't really like there being a segment where people won 95% of the time. So they demanded he reduce people's chance of winning.
Lymon's fix was pretty simple; he switched from 100 envelopes to 3. Now the chance of picking the right envelope for the selected pile was 1-in-3.
Now the sequence was exactly the same as before, except he only burned one envelope instead of 98:
The new producers harangued Lymon about bringing out a portable incinerator just to burn a single envelope, so he changed that. The point of burning the envelopes had effectively been to speed up opening them and showing them empty—the audience understood that Lymon would never burn a non-empty envelope. So if the point was to show them empty, he could just show them empty.
So instead of burning the one envelope, he just showed it was empty.
Lymon had been doing this segment for so long and it had been going for so long he decided it was time to re-examine it with fresh eyes and see if it it could be streamlined.
It was a bit silly to separate the selected envelope into a pile, and then the two remaining envelopes into another pile, and then immediately pull one of the envelopes out of the remaining pile and open it. So he stopped moving the envelopes around at all. He didn't change anything that affected the odds.
Lymon's show got another new set of producers, and they complained this envelope business was pretty simple and stale. They asked if he could make it more visual.
Lymon changed things as follows:
Now the sequence of events was:
The odds didn't change; contestants still had a 2-in-3 chance of winning if they chose the remaining doors instead of the selected door.
The whole thing where you got everything behind both remaining doors, even though one was open and empty, felt pretty silly for Lymon to say, so he phrased things differently to omit mentioning the open, empty door. But this didn't change the odds or the outcome.
The odds were still 2-in-3 for choosing the remaining door, even though the contestant was technically now choosing between two single doors, not between a single door and a pair of doors. This is because according to the official rules they were choosing between the single door and the pair of doors; Lymon just wasn't mentioning it anymore since they were obviously never going to get anything from the empty one, so why confuse things?
Monty Hall saw Steal A Mad Leek on TV while he was developing Let's Make A Deal and decided to copy Lymon's numbered door segment.
Monty puts goats behind the "empty" doors, but since the constestants didn't get to take the goats home, that didn't make any difference—they were still empty in terms of analyzing the odds of winning.
Monty changed the terminology from a first step picking a "selected door" and second step of a final choice between the "selected door" and the "remaining door". In his version, in the first step you chose a door, and in the second step you had the option to "switch" (to the remaining door). This was just a terminology change and didn't make any difference for the odds.
Since neither change made a difference to the odds, the odds were still 2-in-3.
Version M is the traditional Monty Hall problem. The meaningful differences between Version M and version B are:
One could do a much shorter presentation of this by focusing on those differences, instead of the gradual, step-by-step "morph" I did from one to the other, and that might be more useful for some people.