[RPG] the maximum possible chance to critically hit

critical-hitdnd-5eoptimization

What is the highest possible chance to critically hit? This can include content from any official source as well as Unearthed Arcana.

For example the level 15 Champion Fighter can critically hit on a roll as low as an 18. If the character has advantage they roll two dice. The Elven Accuracy racial feat allows you to reroll 1 of the d20s essentially creating double advantage. If you also take the Lucky feat you can roll an additional d20 essentially creating triple advantage.

Note I'm looking for the highest chance for the die roll to result in a crit, NOT factoring in effects that specifically guarantee a crit (e.g. any hit on a paralyzed or unconscious enemy, or features like the rogue's Assassinate feature).

Anything that guarantees a crit including predetermined dice rolls like portent should not be considered

I'm looking for both the methods used and the overall %chance to hit. For reference, the character does not have to be viable in a game; this is pure theory-crafting.

Best Answer

Once a day with a 67.94% chance, sustainably once per turn with a 38.59% chance.

Race: Elf (Any) or Half-elf (Any)
Feats: Elven Accuracy (XGE), Lucky (PHB)
Spells: Find Familiar (PHB)
Magic Items: Luck Blade (DMG)
Classes:

  • Champion Fighter 15
  • Divination Wizard 2
  • Anything 3

From Champion Fighter, you succeed at scoring a critical hit on 3 out of 20 dice rolls (17 chances to not fail).

  1. At the beginning of the day, Portent provides us with 2 banked rolls (effectively 2 rerolls).
  2. Before attacking, our Familiar uses the Help Action to grant us advantage (1 reroll).
  3. Elven Accuracy allows us to roll a third time any time we have advantage (1 reroll).
  4. Lucky allows us to make an additional roll before an attack roll is resolved (1 reroll).
  5. The Luck Blade's Luck feature grants us an additional reroll once per day (1 reroll).
  6. On our attack, we roll once in the first place (1 roll).

In total, we roll a d20 seven times. If any of those rolls would be an 18 or higher, we crit. Alternatively as long as all of them aren't 17 or lower, we crit:

\$ 1 - \left(\frac{17}{20}\right)^7 = 67.94\% \$

This result is very similar to Dale M.'s answer, however it outlines a small improvement: Halfling Luck is conditional on a 1/20 chance of rolling a one per reroll. However often that chance of rerolling is checked, Elven Accuracy grants an additional reroll unconditionally so it is the better feature.

Now, to consider the reliability of this build, we'll look at features that are not expended:

  1. Before attacking, our Familiar uses the Help Action to grant us advantage (1 reroll).
  2. Elven Accuracy allows us to roll a third time any time we have advantage (1 reroll).
  3. On our attack, we roll once in the first place (1 roll).

\$ 1 - \left(\frac{17}{20}\right)^3 = 38.59\% \$

How does this compare to a Halfling with Halfling Luck?

  1. Before attacking, our Familiar uses the Help Action to grant us advantage (1 reroll).
  2. On our attack, we roll once in the first place (1 roll).
  3. If either of our rolls are a natural 1, Halfling Luck lets us reroll (conditional reroll).

\$ \left(1 - \left(\frac{17}{20}\right)^2\right) + \left(1 - \left(\frac{19}{20}\right)^2\right)\left(\frac{17}{20}\right)= 36.04\% \$

For reference, these are the chances depending on how many rerolls you have available for use on the attack:

\$ \begin{array}{|c|c|} \hline \text{Rolls} & \text{Chance} \\ \hline 1 & 15.00\% \\ \hline 2 & 27.75\% \\ \hline 3 & 38.59\% \\ \hline 4 & 47.80\% \\ \hline 5 & 55.63\% \\ \hline 6 & 62.28\% \\ \hline 7 & 67.94\% \\ \hline 8 & 72.75\% \\ \hline 9 & 76.84\% \\ \hline 10 & 80.31\% \\ \hline \end{array} \$

As you can see, with each additional reroll, you start to see diminishing returns. At the end of the day, you CAN spend a casting of Wish on your next turn to reroll the past event to get an 8th reroll, but you'd have only increased your chance of scoring a critical hit by about \$+5\%\$. That casting of Wish could just as easily have been used to cast Sunburst for \$12\text{d}6\$ damage or an 8th Level Scorching Ray for up to \$16\text{d}6\$ (with more chances to crit because Elven Accuracy applies).

Chance to score a critical per turn, at least 67.94%

For this, we're going to assume that you have a source of advantage for the entire turn, which is fairly expected in high level play. Either some Faerie Fire has hit the mark or a target has been knocked prone, Flanking rules are in effect, or you just used a Scroll of Foresight. Doesn't matter where it comes from.

This means that every attack made makes three rolls minimum.

We're also going to assume that the character will be using their bonus action to Two-Weapon Fight for an extra attack per Action.

Further, we're going to assume the character is under the effects of Haste, also a fairly expected boon in high-level play.

  1. At Level 15, Fighters have three main-hand attacks.
  2. Action Surge grants those three attacks an additional time once per turn.
  3. Two-Weapon Fighting grants an additional off-hand attack once per turn.
  4. The extra action from Haste can be used to make an additional main-hand attack per turn.

In total, the character is making eight (8) attacks and each of them is being rolled at least three (3) times.

\$ 1 - \left(\frac{17}{20}\right)^{3 \times 8} = 97.98\% \$

Action Surge is a resource that gets consumed, so in reality, only 5 of those attacks can be expected sustainably.

\$ 1 - \left(\frac{17}{20}\right)^{3 \times 5} = 91.26\% \$

And lets say that you are only granted advantage on that first attack, meaning only \$3 + 4\$ rolls made per turn.

\$ 1 - \left(\frac{17}{20}\right)^{3 + 4} = 67.94\% \$

Hey now, isn't that something? That number looks pretty familiar.