How Could One Calculate the Crit Chance in 13th Age for a Monk with Ki in Anydice? (If It Is At All Possible)

Question

In 13th Age a critical is achieved when you roll a natural 20 on a d20. The Monk class has a few features that can fundamentally change its potential to crit.

First off, a Monk using bare hands to fight is considered to be using Two-Weapon fighting. If a character who is two-weapon fighting rolls a natural 2 on an attack roll, they may reroll but must take the new result.

Second, a Monk has a resource called Ki. As a free action, a Monk can spend a Ki point to adjust their natural attack roll by one, unless that result is a natural 1. This can be a change up or down. This means that a Monk who rolls a 19 can spend a Ki to make it a 20, and a Monk who rolls a 3 can make it a 2 and trigger the re-roll from Two-Weapon Fighting. They can't adjust a natural 1 to be a 2, however.

It should also be noted that while a Monk can take a feat to be able to spend more than 1 Ki per turn, it is impossible for any Monk to spend more than 1 Ki on the same feature more than once in a turn. Therefore a Monk who rolls a 3 can spend 1 Ki to change it to a 2 and get the re-roll, but if the re-roll resulted in a 19 they couldn't then spend an additional Ki point to make it a 20.

So with all that said: Does anyone know IF this can be calculated in Anydice and if so, using what formula?

I barely know how to use Anydice beyond the basics and probability is not my strong suit.

Assumptions

Assume that the player will always reroll on a 2, since the odds of getting a higher result are so high and a 2 is often unlikely to hit. Also assume that we're working within a single turn and don't care about the chances on subsequent turns, which means we can only spend 1 Ki to adjust 1 roll once. The rules wouldn't allow us to spend any Ki to adjust the roll a second time anyhow.

Clarification

Two-Weapon Fighting isn't like advantage in D&D. You roll 1d20 for an attack and should the result of that d20 land on a 2, the player has the choice to re-roll; by doing so, they are replacing the first roll as if it never happened and are stuck with the new roll result, even if it's another 2.

Answer 1

To do a probabilistic branch in AnyDice, you need to use a function with a parameter of type number or sequence, and then feed a die or dice pool to it respectively. The function will then be evaluated over all possible rolls and the results collated into a die. For this problem, we can enumerate the possible cases of the initial roll:

• On a 1, we're stuck with the result.
• On a 2, we reroll, and can then increase the result by 1 provided it is not a 1.
• On a 3, we decrease the result to 2, reroll, and take the result with no adjustment.
• Otherwise, we just increase the result by 1.

Since we only have a single die, a parameter of type n will do fine.

The program:

function: monk R:n {
  if R = 1 {
    result: 1
  }
  if R = 2 {
    result: d{1, 3..21}
  }
  if R = 3 {
    result: d20
  }
  result: R + 1
}

output [monk d20]

The result is a 10.75% chance of scoring a crit.

• 5% from an initial 20.
• 5% from a 19 adjusted to a 20.
• 0.25% from a 2 rerolled to a 20.
• 0.25% from a 2 rerolled to a 19, then adjusted to a 20.
• 0.25% from a 3 adjusted to a 2, then rerolled to a 20.

Answer 2

You can just do the maths

The chance of a crit on the first roll is 0.1 (a 19 or 20). The chance of a clean reroll is 0.05 (a 2) which gives a 0.10 chance of a crit (a 19 or 20) giving a 0.005 chance of a crit this way. Or 0.05 (a 3) followed by 0.05 (a 20) giving 0.0025. So the total chance of a crit is 0.1075.

Answer 3

HighDiceRoller has already offered one solution, but here's another:

function: is ROLL:n a monk crit {
  if ROLL = 2 { result: d20 >= 19 }  \ We can reroll the 2, and if necessary spend Ki to change a rerolled 19 to 20. \
  if ROLL = 3 { result: d20 >= 20 }  \ We can spend Ki to change the 3 to a 2, and then reroll it (but cannot spend Ki again). \ 
  result: ROLL >= 19                 \ Otherwise we crit on 20, and can spend Ki to change a 19 to a 20. \
}

output [is d20 a monk crit]

As the comments in the code note, there are three cases we need to consider:

• If we roll a 2, we can reroll it, and still have a Ki point to spend to upgrade a rerolled 19 to a 20.
• If we roll a 3, we can spend a Ki point to change it to a 2 and the reroll (but can't spend Ki again, so this roll will only crit if it's a natural 20).
• Otherwise we crit on a natural 19 or 20, since we can spend a Ki point to upgrade the 19 to a 20.

Anyway, as the other answers note, the final result is that the crit chance is 10.75%, i.e. just 0.75 %-points higher than the 10% chance of rolling a nat 19 or 20 on the first roll. Basically the rerolls make hardly any difference to the crit chance, because not only is the chance of getting a reroll low in the first place, but the chance of getting a reroll and then critting on the reroll is much smaller yet.