Critical Hit Chance & Hit Damage Explained

Content of this article:

  1. Introduction
  2. The Math behind Critical Strikes
  3. Skill Interactions
  4. Passives, Abilities & Legendary Effects

Introduction

Figure 1: Critical Strikes (yellow)

Critical Strikes are one of the most common and recurring concepts across all ARPGs; it's a normal attack or spell cast dealing extra damage and potentially triggering "on Critical Strike" effects. In Diablo III a Critical Strike is indicated in the User Interface (UI) as yellow damage number (see Figure 1) while white numbers indicate non-crit damage numbers.

While other ARPGs, as an example, differentiate between Spell Critical Strike and Attack Critical Strike, Diablo III does not and simplifies the mechanics at hand significantly. Overall there are only two primary affixes that scale Critical Strikes: Firstly, Critical Hit Chance, which denotes the chance for an individual hit to critically strike a target. Secondly, Critical Hit Damage, which denotes the multiplier a target will be critically striked for.

The guide at hand will go over the mechanics behind Critical Strike and how to evaluate it. Additionally, we'll take a quick look on Damage over Time (DoT) interactions and provide a full list of all passive, abilities and items in the game that provide a buff to Critical Strikes.


The Math behind Critical Strikes

Critical Strikes are fairly simple to analyze. Let's assume your character is hitting for exactly 100 damage each time as a normal hit. With 50% Critical Strike Chance (CHC) you'll have a 50% chance, each time you hit, for each target to be critically hit. Let's further assume, in this scenario, that you have 100% Critical Hit Damage (CHD) equipped; which means if a target is critically hit, it will take 100% additional damage (200 damage). In this coin-flip scenario it's very easy to evaluate what your "average damage hit" is: You have a 50% to hit for 100 damage and a 50% chance to hit for 200 damage meaning average you'll hit for \frac{100 + 200}{2} = 150 damage.

A follow-up question at this point would be though, what happens if you trade-off 5% CHC for 20% CHD; is this trade-off a DPS increase? And if so, by how much? To answer this question we firstly need to take a look at the Expected Value of our damage hits. In our previous coin-flip example we implicitly calculated our expected damage as \frac{100 + 200}{2} = 150 which is a direct consequence of the expected value of our damage D:

    \begin{eqnarray*}E[D] &=& (1-\textrm{CHC}) \cdot 100 + \textrm{CHC} \cdot 100 \cdot (1 + \textrm{CHD})\\ &=& (1-0.5) \cdot 100 + 0.5 \cdot 100 \cdot (1 + 1)\\&=& 0.5 \cdot 100 + 0.5 \cdot 100 \cdot 2 \\&=& 0.5 \cdot (100 + 100 \cdot 2) \\&=& \frac{100 + 200}{2} \\&=& 150\end{eqnarray*}

If we were to make the aforementioned trade-off "5% CHC for 20% CHD" we'd end up at 45% CHC with 120% CHD resulting in an expected damage value of:

    \begin{eqnarray*}E[D] &=& (1-\textrm{CHC}) \cdot 100 + \textrm{CHC} \cdot 100 \cdot (1 + \textrm{CHD})\\&=& (1-0.45) \cdot 100 + 0.45 \cdot 100 \cdot (1 + 1.2)\\&=& 0.55 \cdot 100 + 0.45 \cdot 100 \cdot 2.2 \\&=& 55 + 99 \\&=& 154\end{eqnarray*}

Hence, the trade-off at hand is a \frac{154}{150} \approx 2.67\,\% damage increase. While it's technically correct at this point to the speak of an Expected Damage Value (on the basis of Expected Value), we are in a special case where we only have two random variables: Firstly, \textrm{CHC}, which denotes our chance to critically strike, and secondly 1-\textrm{CHC}, which denotes our chance for a normal hit to occur. With that in mind we can simplify E[D] as follows:

    \begin{eqnarray*}E[D] &=& (1-\textrm{CHC}) \cdot D + \textrm{CHC} \cdot D \cdot (1 + \textrm{CHD})\\&=& D \cdot ((1-\textrm{CHC}) + \textrm{CHC} \cdot (1 + \textrm{CHD})\\&=& D \cdot (1-\textrm{CHC} + \textrm{CHC} + \textrm{CHC} \cdot \textrm{CHD}))\\&=& \underbrace{D}_{\textrm{\tiny{Normal Damage Hit}}} \cdot \underbrace{(1 + \textrm{CHC} \cdot \textrm{CHD})}_{\textrm{\tiny{Crit-Multiplier}}}\\\end{eqnarray*}

The above standing simplification showcases that the mechanic Critical Strike is simply a separate multiplicative damage multiplier. This also means that we can take a much simpler approach when comparing two different CHC/CHD scenarios as previously done; there is technically speaking no reasons for us to go the entire way of calculating two different Expected Damage Values E[D] and afterwards compare their relative performance. We can straight up just compare the Crit-Multiplier of both setups:

    \begin{eqnarray*}\frac{1 + \textrm{CHC}' \cdot \textrm{CHD}'}{1 + \textrm{CHC} \cdot \textrm{CHD}} = \frac{1+0.45 \cdot 1.2}{1+0.5 \cdot 1} = \frac{1.54}{1.5} = 2.67\,\%$ \end{eqnarray*}

The above standing calculation is exactly what the in-game client provides you with, when you for example compare two different rings that have different CHC/CHD values. It simply compares the relative change of the resulting Crit-Multiplier.


Skill Interactions

A Legacy Scorn for Trash Farming MP10

Damage over Time
Critical Hit Chance (CHC) and Critical Hit Damage (CHD) have a special interaction with Damage over Time (DoT) skills. Per definition these skills are not eligible to Critical Strike anymore. This is a result from Diablo III Vanilla, where Barbarians were wielding Skorn, a very high average damage (DMG) weapon, and casting Rend while being surrounded by monsters and therefore receiving a significant amount of Critical Hit Chance via Battle Rage Into the Fray. Back in the days a DoT Skill would continuously apply \textrm{DMG} \cdot (1 + \textrm{CHD}) for every occuring tick, if and only if the first hit did Critical Strike with a chance of \textrm{CHC}.

This got nerfed pretty quickly. Nowadays DoT Skills can not crit anymore whatsoever, yet, they scale with CHC and CHD. Therefore, instead of giving each consecutive tick an individual chance to critically strike the developers decided that each tick of a DoT Skill should be scaled via the Crit-Multiplier 1 + \textrm{CHC} \cdot \textrm{CHD}.

Thorns
Any type of Thorns damage is always Physical, with the exception of the Demon's Skin (2) Bonus and Demon's Hide (2) Bonus that apply Fire Thorns. Thorns damage can never be scaled via Critical Hit Chance (CHC) and Critical Hit Damage (CHD). The only exception to this rule is Envious Blade, where the first damage instance to inflict damage will utilize CHD even if the source is Thorns damage. In all other cases Thorns can not crit itself nor is there a way to somehow apply the Crit-Multiplier 1 + \textrm{CHC} \cdot \textrm{CHD}. This is unfortunately also true for Hack as well as for the Thorns of the Invoker set.


Passives, Abilities & Legendary Effects

With the following Passives, Abilities and Legendary Effects you'll be additionally able to scale your Crit-Multiplier besides the primary affixes "Critical Hit Chance" and "Critical Hit Damage" from your gear:

  • Barbarian
    • Battle Rage specifically the rune Into the Fray
    • Bash Onslaught
    • Call of the Ancients Duty to the Clan
    • Hammer of the Ancients
    • Revenge Best Served Cold
    • Overpower Killing Spree
    • Weapons Master
    • Wrath of the Berserker
  • Crusader
    • Bombardment Annihilate
    • Judgment Resolved
    • Laws of Valor Critical
    • Punish Fury
    • Slash Crush
  • Demon Hunter
    • Archery
    • Caltrops Bait the Trap
    • Multishot Wind Chill
    • Single Out
    • Sharpshooter
    • Vault Action Shot
  • Monk
    • Balance
  • Necromancer
    • Command Golem Ice Golem
    • Corpse Lance Brittle Touch
  • Wizard
    • Conflagration
    • Energy Armor Pinpoint Barrier
    • Frost Nova Deep Freeze
    • Spectral Blade Ice Blades
  • Items
    • Broken Promises
    • Envious Blade
    • Iceblink
    • Shi Mizus Haori

Press F for Witch Doctors as they have not a single class-specific passive, ability nor item that increases Critical Hit Chance/Damage.

Cheers,
—sVr