Luck

Lucky crit rolls visualized in a tree diagram.

Luck is a mechanic that rolls a number twice, with Lucky rolls applying the best result, and Unlucky rolls applying the worst result. Modifiers that state when Damaging refer specifically to damage ranges, and not related mechanics such as accuracy or critical strike chance.

Lucky and Unlucky modifiers affecting the same thing will cancel each other out.

Effectiveness

With percentage rolls, such as critical strike 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:

$\color[RGB]{163,141,109} CritChance_{Lucky} = 2 \times CritChance_{Default} - {CritChance_{Default} } ^ 2$

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

$\color[RGB]{163,141,109} CritChance_{Lucky} = 2 \times 40$

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

Derivation of lucky rolls ▾ [Expand] ▾

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:

$\color[RGB]{163,141,109} B = 1 - A$

(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:

$\color[RGB]{163,141,109} C = B \times A$

(2)

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

$\color[RGB]{163,141,109} \begin{align} CritChance_{Lucky} & = A + C \\ & = A + B \times A \\ & = A + (1 - A) \times A \\ & = 2A - A^2 \end{align}$

(3)

Damage in a Range

Given an integer roll between $\color[RGB]{163,141,109} min$ and $\color[RGB]{163,141,109} max$, the normal expected value is $\color[RGB]{163,141,109} \frac{min+max}{2}$.

A lucky roll has expected value $\color[RGB]{163,141,109} \frac{min}{3} + \frac{2*max}{3} + \frac{max-min}{6*(1+max-min)}$.

Note that the last term in this is always between $\color[RGB]{163,141,109} 0$ and $\color[RGB]{163,141,109} 1/6$ 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 $\color[RGB]{163,141,109} \frac{min+2*max}{3}$.

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 FireballFireballProjectile, Spell, AoE, Fire
Level: (1-20)
Cost: (6-25) Mana
Cast Time: 0.75 sec
Critical Strike Chance: 6.00%
Effectiveness of Added Damage: 240%
Projectile Speed: 1040
Radius: 9Unleashes a ball of fire towards a target which explodes, damaging nearby foes.Per 1% Quality:1Superior2Anomalous3Divergent4Phantasmal1% increased Projectile Speed2% reduced Projectile Speed
3% increased Area of Effect3% increased Effect of non-Damaging Ailments on Enemies
-2% more Damage with Ignite0.25% chance to Debilitate Enemies for 1 second on HitDeals (9-1095) to (14-1643) Fire Damage
(20-39)% chance to Ignite enemies
Projectiles gain Radius as they travel farther, up to +(0-9) RadiusPlace into an item socket of the right colour to gain this skill. Right click to remove from a socket.. The maximum damage is 1643, and minimum damage is 1095, for an average of 1369. The difference between minimum and maximum is 548. If the damage is lucky, you get an average 548/6=91.3 extra damage, for a new average damage of 1369+91.3=1460.3 damage. This is a 6.6% damage boost.

Example 2: You cast a level 20 SparkSparkSpell, Projectile, Duration, Lightning
Level: (1-20)
Cost: (5-21) Mana
Cast Time: 0.65 sec
Critical Strike Chance: 6.00%
Effectiveness of Added Damage: 125%
Projectile Speed: 560Launches unpredictable sparks that move randomly until they hit an enemy or expire.Per 1% Quality:1Superior2Anomalous3Divergent4Phantasmal1% increased Projectile SpeedProjectiles Pierce 0.1 additional TargetsFires 0.1 additional Projectiles2% of Lightning Damage Converted to Chaos DamageDeals (2-29) to (70-1326) Lightning Damage
Base duration is 2 seconds
Fires (4-6) additional Projectiles
(0-19)% increased Projectile SpeedPlace into an item socket of the right colour to gain this skill. Right click to remove from a socket.. 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 $\color[RGB]{163,141,109} 0$ then we get a $\color[RGB]{163,141,109} 33$ 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
Coward's ChainsCoward's Chains Chain BeltRequires Level 22+(9-20) to maximum Energy Shield+(10-15) to all Attributes (5-10)% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are Cursed with Vulnerability You count as on Full Life while you are Cursed with Vulnerability You are Cursed with Vulnerability, with 40% increased EffectAs a gift or as punishment, the Empire will have your blood. This item can be transformed on the Altar of Sacrifice along with Vial of Consequence	Chain Belt	22	+(9-20) to maximum Energy Shield+(10-15) to all Attributes (5-10)% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are Cursed with Vulnerability You count as on Full Life while you are Cursed with Vulnerability You are Cursed with Vulnerability, with 40% increased Effect
Relic of the PactRelic of the Pact Spiraled WandWand Quality: +20% Physical Damage: 14.4-44.4 Critical Strike Chance: 7.00% Attacks per Second: 1.30 Weapon Range: 120Requires Level 24, 83 Int(15-19)% increased Spell DamageGrants Level 1 Blood Sacrament Skill Your Critical Strike Chance is Lucky while on Low LifeCrush your enemies with your essence, so that you may drink of theirs.	Spiraled Wand	24	(15-19)% increased Spell DamageGrants Level 1 Blood Sacrament Skill Your Critical Strike Chance is Lucky while on Low Life
Perquil's ToePerquil's Toe Gold AmuletRequires Level 29(12-20)% increased Rarity of Items found+(40-50) to Dexterity (5-8)% increased Movement Speed Lightning Damage from Enemies Hitting you is Lucky Nearby Allies' Damage with Hits is LuckyHe was a lot luckier before he lost this.	Gold Amulet	29	(12-20)% increased Rarity of Items found+(40-50) to Dexterity (5-8)% increased Movement Speed Lightning Damage from Enemies Hitting you is Lucky Nearby Allies' Damage with Hits is Lucky
Lori's LanternLori's Lantern Prismatic RingRequires Level 30+(8-10)% to all Elemental Resistances+10% to all Elemental Resistances (6-8)% increased Movement Speed when on Low Life 31% increased Light Radius +(20-25)% to Chaos Resistance when on Low Life Damage from Enemies Hitting you is Unlucky while you are on Low LifeBy its light, Lori led her young charges to safety and bound up their wounds. Her sword arm was iron, yet her heart gentle and proof against hate.	Prismatic Ring	30	+(8-10)% to all Elemental Resistances+10% to all Elemental Resistances (6-8)% increased Movement Speed when on Low Life 31% increased Light Radius +(20-25)% to Chaos Resistance when on Low Life Damage from Enemies Hitting you is Unlucky while you are on Low Life
Fox's FortuneFox's Fortune Wild LeatherQuality: +20% Evasion: (1572-2038) Movement Speed: -3%Requires Level 54, 73 Dex+(20-30) to Dexterity Adds 5 to 12 Physical Damage to Attacks +1500 to Evasion Rating while on Full Life +(400-500) to Evasion Rating (120-150)% increased Evasion Rating 10% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are on Full LifeFollow your most basic instincts. The rest will come naturally.	Wild Leather	54	+(20-30) to Dexterity Adds 5 to 12 Physical Damage to Attacks +1500 to Evasion Rating while on Full Life +(400-500) to Evasion Rating (120-150)% increased Evasion Rating 10% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are on Full Life
The FulcrumThe Fulcrum Ezomyte StaffWarstaff Quality: +20% Physical Damage: (137.8-159) to (416-480) Critical Strike Chance: 7.30% Attacks per Second: 1.25 Weapon Range: 13Requires Level 60, 113 Str, 113 Int+20% Chance to Block Attack Damage while wielding a Staff(140-180)% increased Physical Damage (0-50)% of Physical Damage Converted to Fire Damage (0-50)% of Physical Damage Converted to Cold Damage (0-50)% of Physical Damage Converted to Lightning Damage Elemental Ailments you inflict are Reflected to you Elemental Damage with Hits is Lucky while you are Shocked Damage Penetrates (8-10)% Elemental Resistances while you are Chilled Gain (30-40)% of Physical Damage as Extra Damage of a random Element while you are IgnitedTo stand at the confluence of the elements, the master must achieve perfect balance.	Ezomyte Staff	60	+20% Chance to Block Attack Damage while wielding a Staff(140-180)% increased Physical Damage (0-50)% of Physical Damage Converted to Fire Damage (0-50)% of Physical Damage Converted to Cold Damage (0-50)% of Physical Damage Converted to Lightning Damage Elemental Ailments you inflict are Reflected to you Elemental Damage with Hits is Lucky while you are Shocked Damage Penetrates (8-10)% Elemental Resistances while you are Chilled Gain (30-40)% of Physical Damage as Extra Damage of a random Element while you are Ignited
The Iron FortressThe Iron Fortress Crusader PlateQuality: +20% Armour: (1202-1322) Movement Speed: -5%Requires Level 68, 160 Str+1% Chance to Block Attack Damage per 50 Strength Chance to Block Spell Damage is Unlucky +(60-120) to Strength (80-100)% increased Armour 10% reduced Movement Speed Strength's Damage Bonus instead grants 3% increased Melee Physical Damage per 10 Strength"Stand steadfast in the face of destruction. The Council cannot afford to falter." - Mauritius, the Iron Heart	Crusader Plate	68	+1% Chance to Block Attack Damage per 50 Strength Chance to Block Spell Damage is Unlucky +(60-120) to Strength (80-100)% increased Armour 10% reduced Movement Speed Strength's Damage Bonus instead grants 3% increased Melee Physical Damage per 10 Strength
Viridi's VeilViridi's Veil Praetor CrownQuality: +20% Armour: 193 Energy Shield: 57Requires Level 68, 62 Str, 91 Int+(1–2) to Level of Socketed Gems (120–160)% increased Armour and Energy Shield +(15–25)% to all Elemental Resistances Damage of Enemies Hitting you is Unlucky while you have a Magic Ring Equipped You are Hexproof if you have a Magic Ring in right slot Take no Extra Damage from Critical Strikes if you have a Magic Ring in left slot"Her sisters eternally fought for that shining apex in the skies, but Viridi instead found strength in humility." - Azmerian Creation Myth	Praetor Crown	68	+(1–2) to Level of Socketed Gems (120–160)% increased Armour and Energy Shield +(15–25)% to all Elemental Resistances Damage of Enemies Hitting you is Unlucky while you have a Magic Ring Equipped You are Hexproof if you have a Magic Ring in right slot Take no Extra Damage from Critical Strikes if you have a Magic Ring in left slot

Related skill gems

The following skill gems are related to Luck:

Skill gem
Vaal ArcVaal ArcVaal, Spell, Chaining, Lightning, Duration Level: (1-20) Souls Per Use: 25 Can Store 1 Use(s) Soul Gain Prevention: 0 sec Cast Time: 0.80 sec Critical Strike Chance: 5.00% Effectiveness of Added Damage: 180%Requires Level 12A shocking arc of lightning stretches from the caster to a targeted enemy and chains to other nearby enemies. Each time the beam chains it will also chain simultaneously to a second enemy, but no enemy can be hit twice by the beams. Also grants a buff making you lucky when damaging enemies with Arc for a short duration.Per 1% Quality:1Superior1.5% increased Duration of Lightning AilmentsBase duration is 4 seconds Deals (38-924) to (64-1539) Lightning Damage 100% chance to Shock enemies Chains +(5-9) Times 100% increased Effect of Lightning Ailments 100% increased Duration of Lightning Ailments 15% more Damage for each remaining Chain Modifiers to Buff Duration also apply to this Skill's Soul Gain Prevention CorruptedPlace into an item socket of the right colour to gain this skill. Right click to remove from a socket.	N/A	N/A		12

Related passive skills

Passive skills

The following passive skills are related to Luck:

Name	Stats
Supercharge	Lightning Damage with Non-Critical Strikes is Lucky [1]

Keystone passive skills

The following keystone passive skills are related to Luck:

Name	Stats
Dance with Death	Can't use Helmets Your Critical Strike Chance is Lucky Your Damage with Critical Strikes is Lucky Enemies' Damage with Critical Strikes against you is Lucky

Related modifiers

The following modifiers are related to Luck:

Name	Generation Type		Stats	Spawn Weighting
of the Order	Suffix	50	Your Critical Strike Chance is Lucky while Focussed	belt 0 default 0
of the Order	Suffix	72	Your Critical Strike Chance is Lucky while Focussed	belt 2000 default 0
of Miring	Suffix	46	Player chance to Dodge is Unlucky Monsters have 20% increased Accuracy Rating	default 800
of Miring	Suffix	68	Player chance to Dodge is Unlucky Monsters have 30% increased Accuracy Rating	default 800
of Miring	Suffix	73	Player chance to Dodge is Unlucky Monsters have 40% increased Accuracy Rating	default 800
of Miring	Suffix	78	Player chance to Dodge is Unlucky Monsters have 50% increased Accuracy Rating	default 800