A number of video games choose to make the gambler's fallacy true -- if you roll X once, you are less likely to roll it again in the immediate future. This limits runs of both good and bad luck.

This is technically possible in an RPG, by replacing dice with cards or chits -- once you've drawn a given value, getting it again is less likely (or even impossible) because that particular card/chit is no longer in the deck, and this will last until you return the discards to the pool. For D&D, the easiest way to do this would probably be removing the face cards from a deck of cards, then deciding that one color (either red or black) is +10. This gives you a 40 card deck with two of each value 1-20 -- the odds of a given value start at 2/40, but goes up as the deck size goes down, and goes down based on the number of times it has already happened.

Has anyone experimented with something like this? Does it seem like a good idea?

 I've thought about using a deck of cards as a RNG, most often in the context of the Tarokka Deck from Curse of Strahd. It's certainly interesting, but I've never implemented it because anything other than the D20 just doesn't feel like D&D to me.

However, the card deck would at a certain point make rolls very prescriptive. If you have already spent all your high cards, maybe you don't attempt that high DC check to save an ally. If you know you've still got your 20s in the deck, maybe you go crit fishing against the big bad. Also, when do you reshuffle? If it's when the deck is out, you'll eventually get a sense for what cards are in the deck, and you'll always know what that last roll will be when there's only 1 card left in the deck.

All the questions above are solved by simply reshuffling a card into the deck after you draw it, but that makes the point of using a deck of cards moot. It just becomes harder to read dice that you have to shuffle after every roll.

What could be interesting is giving characters the ability to take out certain cards after they draw them. For example, did your fighter draw that Nat 1 card? Well, they can use their thrice-a-day special ability to take that card out, effectively boosting their odds across the board. But I'd imagine such a mechanic would be restrained to a campaign where all of the characters 'rolled' like that and the overall power level was increased slightly to compensate.

Personally, I like to tilt the luck scales of D&D by increasing bonuses at a rate faster than I increase DCs. I'd give heroes +2 Armor and bonuses that give them more + To Hit, all the while the common bandits or orcs don't get the luxury. It makes the game more heroic, which I like (your mileage may vary).

Typically games that use card pools are designed to reshuffle part way through a deck, such as by having a card which, when drawn, triggers a reshuffle. In practice I suspect that would be unnecessary for D&D games, how often do you make 40 rolls in a session to start with?

Well, for my part, if I am not making at least 40 rolls in a session, I am not scaring the hell out of my players enough. But, yeah.

it is really fuzzy, but I think we tried something like that in the late 90's,  but couldn't give you specifics, and we abandoned it fairly quickly as it seemed to  just feel wrong.

That was 20 years ago, though, and we are different people than we were now.

I think it could work, but personally I prefer the players to have their own luck normalization tools - things like Inspiration or reroll abilities like Lucky or Silvery Barbs, or additive bonuses like Bardic Inspiration and Flash of Genius, or skipping a roll entirely to take 10/take 20 etc - that they can choose to use when the dice might go against them, rather than a system like this that gives the DM yet another thing they need to keep track of.

These abilities put an interesting choice in the hands of the players - "do I use my "save-the-day" ability on this bad roll, or save it for one that might be even worse later in the day?" - and give them the tools to fight the capricious randomness of the dice without eliminating it entirely. And if all else fails, you get to control the results of most bad rolls even without fudging anything. That's perhaps the biggest advantage of a DM over a video game - you don't need to code in every possible outcome or permutation of a check ahead of time, you can decide on the fly whether your players have been screwed enough by their dice and need a reprieve, and turn that natural 4 they just rolled into "success at a cost" rather than abject failure.

I've seen DMs give Inspiration to anyone who rolls a Nat-1 in combat - I don't exactly like how it dilutes Inspiration as something that people get fairly often, but then it's also nice for bad luck protection and making it less likely that someone goes thru a whole combat without contributing anything productive

Inspiration for a Nat 1 was one of the rules proposed for One D&D. I never played at a table that used that, but it seems like a great idea. I hope they keep it.

The reminder that inspo is a thing kicked the back of my head and so I should note that I do have a current mechanicsm for it.

Not only do we use inspo, I switched around some of the related stuff.

So, I use milestones for my game, but I wanted them to represent effort, so it isn't the kind of milestone system that says "ok, now everyone is x level". No, it is Hero Points that you need x number of to achieve (so, essentially XP with a different face, and I use both systems).

However, a character can use both Hero Points and Inspo on the same roll -- and when I say they use Hero points, I mean they use them -- they sacrifice a bit of their ability to advance to achieve a goal.

When they spend their Hero points to go up a level, they are spending them, and this is also a way of spending them. Inspo is cheap, but hero points are hard, and I give them out for only a few things, but one person earns one at the end of each session as well.

Hero points are given for, well, heroic actions, really creative problem solving, flabbergasting the DM, moving the story forward, and general outstanding play. They give me the freedom to not have to railroad folks (they can also earn Xp, but the two systems don't trade well). To give you an idea of how rare they are in a campaign that takes someone from 1st to 20th level, it is only 270 points total. Lst campaign I think I gave out a grand total of around 290 per character over three years? Maybe 300?

But, to stay on topic, the point of all of that is to show that like inspo, it is a way to recover from nad rolls, to change bad rolls, and to do so without the penalties and bonuses that I often use widely within my games. Inspo is given out much more freely, and is like a "Hero point light" for us.

So I suppose I do have a normalization mechanic. My biggest problem is that it is one sided: it only applies to PCs.