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The SRD for the wish spell states:

Finally, there is a 33 percent chance that you are unable to cast wish ever again if you suffer this stress.

I'm trying to figure out how to handle this limitation. The rest of the question assumes that the caster is casting wish for purposes other than casting a level 1-8 spell.

My grasp of statistics is OK. My DM's? Less so.

How I expect it works: every time I cast wish there is 33% chance I may never cast it again. But I want to cover the naïve questions I expect that my DM is going to have. For instance:

1. I want to know if a caster is bound to not be able to cast wish anymore, meaning that in the best case after 33 casts without suffering the stress, the caster will definitely suffer it on the 34th cast.




2. I would like to know who or what decides that 33 percent is achieved. Does a die have to be rolled? Must another random mechanism be used? And who triggers it: the DM or the player?




3. If the DM/player wants to roll, can the roll be a d6, or must it be a d100? The spell says "33.0" percent, right? Not 1/3 (or 33.333... percent). So a d6 can't be used, can it?




4. Can feats, abilities or spells altering the roll be used?

I first want to know if a caster is bound to not be able to cast wish anymore, meaning that in the best case after 33 casts without suffering the stress, the caster will definitely suffer it on the 34th cast.

If the caster fails their "stress test" then yes, they are bound do not be able to cast wish anymore. But the "33 tries, then fail on the 34th" isn't how probability works. (This is; the short of it is that "best" case is that the stress test never fails, but that 70% of casters will experience that failure by their third casting.)

Second, I would like to know who or what decides that 33 percent is achieved. Does a dice have to be rolled? Must another random mechanism be used? And who triggers it: the DM or the player?

You are free to execute the mechanics of D&D however you like. Most people use dice as their random number generators. Many use high-quality computerized PRNGs that feature animations of dice in their readout. You can flip coins, draw cards, pebbles from an urn..., or anything that your group agrees upon. For standardization the authors assume everyone is using dice and they use dice notation to express quantities.

The player triggers it, by casting wish. Whether the player or the GM rolls (or otherwise initiates random number generation) is up to your table's conventions.

Finally, if the DM/player wants to roll, can the roll be a d6, or must it be a d100? The spell says "33.0" percent, right? Not 1/3 (or 33.333... percent). So a d6 can't be used, can't it?

The spell indeed says 33%, so you're right that "1-2 on a d6" isn't quite RAW. It's off by 1-in-300. Your table will have to decide whether that level of error is tolerable if someone wants to use a d6 (or d12, even!) in place of percentile dice.

Also, can feats, abilities or spells altering the roll be used?

It's going to depend on the feat/ability/spell. But it's almost certainly "no," given the wording of many of those abilities. Note that this roll isn't an attack, check, or save, which Halfling Luck or [dis]advantage would require. The Lucky feat or divination wizard's Portent require a d20 roll. Bless or bane or guidance... none of those would apply. The point being: the source of a die modification should tell you enough as to whether it'd apply to this roll.

There's clearly some misunderstanding here, but I'm not sure if it's a misunderstanding of the game rules or just how probability works in general. You may be thinking in terms of the Gambler's Fallacy -- if you flip a totally fair coin and it comes up heads six times in a row, the chance of getting heads on the next flip is still 50%. It's not "due" for a tails or anything. (If you flipped heads six times in a row on an actual coin, there's a higher chance that the next flip is also heads, because it's increasingly likely that the coin isn't actually fair -- the weighting of the coin or the way you flip it might be causing it to prefer heads -- but the difference would have to be pretty large to have an impact that dramatic.)

Generally, when the game rules refer to a percent chance, you should read that as rolling d100 ('percentile'). The event happens if you roll equal to or lower than the stated number. You can use any fair method of generating numbers, but d100 is usually the simplest way. In this case I would tend to have the caster roll for themself, but the DM could certainly roll instead if they wanted to, since the spell doesn't actually specify.

Each time you cast wish to do one of the 'other' effects, you roll 1d100, and if you get a 33 or lower, you lose the ability to use wish ever again. In theory, you could use wish hundreds of times, as long as you continue to roll over 33 each time. There's no guarantee of failure on roll #34, either by the game rules or by the physics of how dice operate -- so no, the caster isn't bound to eventually roll poorly and lose wish for good.

Only one caster in six will get past their fifth use of wish without losing it forever, though, and only one in a hundred will pass the tenth roll successfully. Getting past the 32nd roll is a trick only one caster in 600,000 can pull off, and managing to wish fifty times in a row without losing the spell is a one-in-a-billion chance. But it can still happen...

As to the other questions, 1-2 on 1d6 is pretty close, but would have a minor deviation of a third of a percent, as you identified, so I wouldn't do that, myself; I'd roll an actual percentile. That said, the difference is so small it probably doesn't matter.

I don't know of any spells, feats, or abilities that could change a percentile roll that's part of a specific spell; did you have something specific in mind? Mostly roll adjustment abilities like Lucky say what they can affect, like 'attack rolls, saving throws, or ability checks', and this roll is none of those. There are a few abilities that can change a percentile roll, but the ability applies to a single, specific kind of roll (like the Wild Magic Sorcerer eventually getting to roll twice on the Surge chart and pick one of the two results). They wouldn't apply to a wish.

You aren't guaranteed to eventually lose wish, just exceedingly likely.

The odds of losing wish to any given major use is 33% or 0.33 regardless of the number you cast previously. You are never guaranteed to lose wish, and your tenth wish is no more risky than the first.

However, the cumulative odds of sucessfully casting spells means that you are overwhelmingly likely to lose wish if you cast ten times, because the odds of at least one failure in ten tries is very high.

We calculate the odds as the chance of the event not happening (67% or 0.67), to the power of the number of rolls, and take that result away from 1 (or 100%). Our chance to lose wish given a certain number of rolls is as follows:

1. 33.00%


2. 55.11%


3. 69.92%


4. 79.85%


5. 86.50%


6. 90.95%


7. 93.94%


8. 95.94%


9. 97.28%


10. 98.18%

Your eleventh casting would still have only a 33% chance to lose wish, but there's a 98.18% chance that you don't even get an eleventh casting because you already lost it to one of the ten previous 33% chances.

To answer your second question, the wish rules don't actually specify what method is to be used to determine the chance to lose wish, and therefore it's technically entirely up to the DM. In fact, as per PHB p.6, wish doesn't explicitly require you to roll dice, because it doesn't specifically tell you to:

When you need to roll dice, the rules tell you how many dice to roll of a certain type, as well as what modifiers to add.

However, we have two guidelines:

• The only method given in the rules to calculate percentages is percentile dice (PHB p.6). The most logical method is that 01-33 is a loss, and 34-00 is no loss, although this isn't specifically stated in the rules. A d6 could be rolled but it's not precisely 33% chance and it's up to the DM to allow that; it's not in the rules.


• The player traditionally rolls their own dice for spell effects, but this isn't included in the rules.

Possible mitigations

The rules provide a few possible methods by which a character who loses their ability to cast wish might regain it:

• Another character can cast wish, and wish for the first character to regain their ability to cast wish (see Can you use Wish to allow yourself to cast Wish without risks?)


• A cleric can pray for divine intervention


• A character may meet a deity in person, and ask that way


• The DM may invent some method to regain wish in their campaign world, or may just ad-hoc forgive the character or change the wish rule


• The DM allows his players to generate new characters of the average party level; the player retires Foplo, the 19th level wizard, and creates a new character, Boplo, identical in every way except he has never failed to cast wish
  

1. It’s not how many times you cast it, but everytime you cast it, that beyond spell duplication you get stressed.




2. I don’t know of any rule about who rolls, I would go with you roll being you are casting the spell whilst taking the risk.




3. Either roller should use a percentile dice roll which is explained below. As mentioned in a comment probability isn’t doesn’t equate to %.

% Chance is based off of a d100, which is an actual die, although they are hardly used, rolls like a golf ball, and are hard to read. You can follow the quote below for rolling for percentile.

As far as Wish is concerned, casting the spell other than just lower spell duplcation (8th or lower) you 1/3rd of dice results or 33% of not being able to ever cast it again.

Wish is the mightiest spell a mortal creature can cast.

It’s just that powerful!

Game Dice

Percentile dice, or d100, work a little differently. You generate a number between 1 and 100 by rolling two different ten-sided dice numbered from 0 to 9. One die (designated before you roll) gives the tens digit, and the other gives the ones digit. If you roll a 7 and a 1, for example, the number rolled is 71. Two 0s represent 100. Some ten-sided dice are numbered in tens (00, 10, 20, and so on), making it easier to distinguish the tens digit from the ones digit. In this case, a roll of 70 and 1 is 71, and 00 and 0 is 100.

Roll result 33 or less = failure to overcome the stress you suffer from casting the spell.