Luck

Lucky crit rolls visualized in a tree diagram.

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Luck is a mechanic that rolls an outcome or a number in a range twice, with Lucky rolls applying the best result, and Unlucky rolls applying the worst result.

Modifiers that specify Damage or when Damaging and luck refer specifically to damage ranges after any increased/more modifiers (or "tooltip damage range"), not to the base damage of the weapon/skill. These stats do not affect other mechanics like chance to hit/accuracy or critical hit chance. Additionally, these stats only apply to hit damage, and have no effect on the damage roll of damage over time from damaging ailments, which roll entirely independently of the hit damage's roll.

Lucky and Unlucky modifiers affecting the same stat or mechanic will cancel each other out.

Effectiveness

With percentage rolls, such as critical hit chance, lucky/unlucky rolls will be up to a 100% more/less modifier, changing linearly towards 0% as critical chance increases. As for damage rolls, the average damage from the roll will be at up to 33% more or less, when the minimum damage is close to zero, but will have a much weaker effect when the minimum damage is closer to the maximum.

Calculations

Binary Rolls

Either roll can make the hit critical, so your critical strike chance on the solid line will be increased to the dash-dotted line on lucky rolls.

More specifically, the lucky critical strike chance can be calculated from the following formula:

$\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}CritChance_{Lucky}=2\times CritChance_{Default}-{CritChance_{Default}}^{2}$

For example, the lucky critical strike chance for an default critical strike chance of 40% could be calculated as follows:

$\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}CritChance_{Lucky}=2\times 40\%-40\%^{2}=64\%$

It should be noted that these calculations neglect the effect of accuracy on critical strikes.

Derivation of lucky rolls

This is a problem that can be solved with conditional probability.

Let A be the default critical strike chance on the first roll, then the chance to NOT crit on the first roll is:

$$\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}B=1-A$$

(1)

If the roll failed the die will be cast a second time with the same probabilities. The chance to crit on the second roll is 1 multiplied with A:

$$\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}C=B\times A$$

(2)

The total probability to get a lucky critical strike is the sum of A and 2.

$\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\begin{aligned}CritChance_{Lucky}&=A+C\\&=A+B\times A\\&=A+(1-A)\times A\\&=2\times A-A^{2}\end{aligned}}$

Damage in a Range

Given an integer roll between min and max, the normal expected value is $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\frac {min+max}{2}}$.

A lucky roll has expected value $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\frac {min}{3}}+{\frac {2*max}{3}}+{\frac {max-min}{6*(1+max-min)}}$.

Note that the last term in this is always between 0 and 1/6 and therefore is negligible (it is a byproduct of the fact our uniform variable only takes values over the integers.) This simplifies the formula to $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\frac {min+2*max}{3}}$.

The average damage is effectively nudged closer toward the max damage, and further from the minimum damage. The damage boost, on average, is a sixth of the difference between minimum and maximum damage.

Example 1: You shoot a level 20 Frost BombFrost BombSpell, AoE, Cold, Duration, Orb, Remnant
Tier: 1
Level: (1-20)
Cost: (9-99) Mana
Cooldown Time: 6.00 sec
Can Store 1 Use(s)
Cast Time: 0.80 sec
Critical Strike Chance: 13.00%Create a pulsing Orb of frost. Each pulse inflicts Elemental Exposure on nearby enemies. When the Orb's duration ends, it Detonates, dealing Cold damage to surrounding enemies and leaving behind a Cold Infusion.Deals (7-238) to (11-358) Cold Damage
Detonation Time is 4 seconds
Chills Enemies as though dealing 300% more damage
Initially applies 20% Elemental Exposure and 2% Exposure per pulse, up to a maximum of 50%
Pulse and explosion radius are (2.4-3.1) metres
Limit 1 Frost Bomb
Exposure duration is 8 seconds

Additional Effects From 1-20% Quality:
Initially applies 0% Elemental Exposure and 0% Exposure per pulse, up to a maximum of (0.5-10)%Skills can be managed in the Skills Panel.. The minimum damage is 578, and the maximum damage is 867, for an average of 722.5. The difference between minimum and maximum is 289. If the damage is lucky, you get an average 289/6=48.1 extra damage, for a new average damage of 722.5+48.1=770.6 damage. This is a 6.6% damage boost.

Example 2: You cast a level 20 SparkSparkBarrageable, Spell, Projectile, Lightning, Duration
Tier: 1
Level: (1-20)
Cost: (6-64) Mana
Cast Time: 0.70 sec
Critical Strike Chance: 9.00%Launches a spray of sparking Projectiles that travel erratically along the ground until they hit an enemy or expire.Deals (1–12) to (8–229) Lightning Damage
(0–19)% increased Projectile Speed
Fires (6–13) Projectiles
Projectile duration is 2 seconds

Additional Effects From 1-20% Quality:
Fires (0–4) ProjectilesSkills can be managed in the Skills Panel.. The maximum damage is 1198, and minimum damage is 63, for an average of 630.5. The difference between minimum and maximum is 1135. If the damage is lucky, you get an average 1135/6=189.2 extra damage, for a new average damage of 189.2+630.5=819.6 damage. This is a 30% damage boost.

How much does lucky improve our expected value - if our minimum damage is 0 then we get a 33% more multiplier. As the minimum damage increases our percentage increase decreases.

Related items

Unique items

The following unique items are related to Luck:

Item	Base item		Stats
The Barrow DwellerThe Barrow Dweller Rogue ArmourArmour: 53 Evasion: 43 Movement Speed: -4.0%Requires Level 11, 14 Str, 14 Dex(60-100)% increased Armour and Evasion +(-20--10)% to Fire Resistance +50% to Cold Resistance Damage of Enemies Hitting you is Unlucky while you are on Low Life 50% chance to Avoid Death from HitsIn the mists they dwell, forever hungry, forever cold.	Rogue Armour	11	(60-100)% increased Armour and Evasion +(-20--10)% to Fire Resistance +50% to Cold Resistance Damage of Enemies Hitting you is Unlucky while you are on Low Life 50% chance to Avoid Death from Hits
The Black CatThe Black Cat Grounding CharmLasts 3.00 Seconds Consumes 30 of 40 Charges on use Grants Immunity to ShockRequires Level 32Used when you become Shocked(10–20)% increased Duration Lightning Damage of Enemies Hitting you is Unlucky during effectThe most beloved member of every Brinerot crew is the one that refuses to do any actual work.Used automatically when condition is met. Can only hold charges while in belt. Refill at Wells or by killing monsters.	Grounding Charm	32	Used when you become Shocked(10–20)% increased Duration Lightning Damage of Enemies Hitting you is Unlucky during effect
The BurrowerThe Burrower Topaz RingRequires Level 32+(20-30)% to Lightning Resistance-10% to Cold Resistance +(20-30)% to Lightning Resistance (30-50)% increased Mana Regeneration Rate Lightning Damage of Enemies Hitting you is UnluckyIt coils deeper and deeper It slithers between thoughts It lies beneath the valley It lies in our minds	Topaz Ring	32	+(20-30)% to Lightning Resistance-10% to Cold Resistance +(20-30)% to Lightning Resistance (30-50)% increased Mana Regeneration Rate Lightning Damage of Enemies Hitting you is Unlucky
SvalinnSvalinn Crucible Tower ShieldShield Chance to Block: 26% Armour: 112 Movement Speed: -3.0%Requires Level 72, 66 StrGrants Skill: Raise Shield Grants Skill: Cast on Block(150-200)% increased Armour You take (15-20)% of damage from Blocked hits Block chance is LuckyThe priests found the Great Shield the night it fell to Middengard, but it was the smiths who delved into the secrets it held.	Crucible Tower Shield	72	(150-200)% increased Armour You take (15-20)% of damage from Blocked hits Block chance is Lucky

Related passive skills

Passive skills

The following passive skills are related to Luck:

Ascendancy passive skills

The following Ascendancy passive skills are related to Luck:

Keystone passive skills

The following keystone passive skills are related to Luck:

Related modifiers

The following modifiers are related to Luck: No results found