Odds

Try it here: 2d6

const spec = parseNotation("4d6dl1")!;
expectedTotal(spec);        // 12.2445987654321
spreadOf(spec);             // 2.85 (standard deviation)
mostLikely(spec);           // [13]
chanceExactly(spec, 18);    // 0.0162 (21 in 1296)
chanceAtLeast(spec, 15);    // the chance of 15 or more
luckOf(spec, 12);           // 0.448: a 12 beats 44.8% of rolls, a tie counted as half
distributionOf(spec);       // { min: 3, max: 18, probabilities: [ … ] }

Nothing is simulated and nothing is left out:

Dice	How the odds are found
Plain dice, any size, and Fate dice	Every outcome counted in whole numbers (BigInt), however many there are
Keep one (advantage)	Counted in whole numbers by the closed form: the highest is at most k when every die is
Keep or drop several	The sum of the highest n, dealt out value by value over the pool
Rerolls, once or until clear	Each face's chance of being the one left standing, then as above
Exploding dice	Each chain's chance up to the limit, then summed over the dice
Loaded and custom dice	Counted in whole numbers like fair dice, each face standing for as many outcomes as it weighs
Several kinds of dice	Each kind as above, then every pair of their totals: a convolution
Dice held	The dice still to roll as above, moved up by the held faces

Counts. Ten d1000 has 10^30 outcomes, far more than a JavaScript number holds exactly, so plain dice and advantage are counted as BigInt and only turned into a probability at the end. exactCounts hands the whole numbers out:

exactCounts(parseNotation("8d6")!);
// { min: 8, outcomes: 1679616n, counts: [1n, 8n, 36n, … ] }   135954n of them total 28
exactCounts(parseNotation("1d20+1d4")!); // { min: 2, outcomes: 80n, counts: [1n, 2n, 3n, 4n, 4n, … ] }
exactCounts(parseNotation("4d6dl1")!);   // null: see below

Probabilities. Rerolled and exploding dice, and several dice kept from a pool, do not have equally likely outcomes to count, so their odds are worked out as probabilities, right to the last digits a number holds (about fifteen). exactCounts returns null for them and never a guess.

Exploding dice and rerolls until clear could go on for ever in theory. Here a die stops after 10 explosions, and after 10 rerolls, in the roll and in the odds alike, so the odds are exactly those of the dice as thrown and they sum to 1. Nothing is approximated and no remainder is left over. A d6 exploding, or rerolling its 1s, reaches that limit once in 60 million dice; a reroll until clear of half a die's faces, the most allowed, reaches it once in a thousand.

Speed. The slowest roll the package accepts, 10d1000kh9, takes about a tenth of a second the first time and nothing after: each spec's odds are remembered. Large mixed pools (5d1000+5d999) are put together as probabilities, and exactCounts returns null for them.

Odds ​

Odds