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.

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

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.

Damage in a Range

Given an integer roll between $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}min$ and $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}max$, the normal expected value is $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\frac {min+max}{2}}$. A lucky roll has expected value Failed to parse (syntax error): \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 $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}0$ and $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}1/6$ and therefore is negligible (it is a byproduct of the fact our uniform variable only takes values over the integers.)

How much does lucky improve our expected value - if our minimum damage is $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}0$ then we get a $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}33\%$ more multiplier. As the minimum damage increases our percentage increase decreases.

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

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

The total probability to get a lucky critical strike is the sum of 1 and 2. $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\begin{aligned}CritChance_{Lucky}&=A+C\\&=A+B\times A\\&=A+(1-A)\times A\\&=2\times A-A^{2}\end{aligned}}$

Related items

Base items

Item	Item Class		Buff Effects
Observer Diamond Flask EffectObserver Diamond Flask EffectFlask SkinActivating your Diamond Flask summons a tentacled abomination that fires its eye lasers at enemies you critically strike while the flask is active.	Cosmetic Item	1	N/A
Diamond FlaskDiamond FlaskLasts 6.00 Seconds Consumes 20 of 40 Charges on use 100% increased Global Critical Strike ChanceRequires Level 27Right click to drink. Can only hold charges while in belt. Refills as you kill monsters.	Utility Flask	27	100% increased Global Critical Strike Chance

Unique items

Item	Item Class		Stats
Coward's ChainsCoward's Chains Chain BeltRequires Level 22+(9-20) to maximum Energy Shield+(15-20) to all Attributes (5-10)% increased Movement Speed Damage of 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 VulnerabilityAs 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	Belt	22	+(9-20) to maximum Energy Shield+(15-20) to all Attributes (5-10)% increased Movement Speed Damage of 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
FoxshadeFoxshade Wild LeatherEvasion: (382-520) Movement Speed: -3%Requires Level 25, 73 Dex+(20-30) to Dexterity +1000 to Evasion Rating while on Full Life (50-70)% increased Evasion Rating 30% increased Movement Speed when on Full Life Damage of Enemies Hitting you is Unlucky while you are on Full LifeTo catch an animal, think like an animal.	Body Armour	25	+(20-30) to Dexterity +1000 to Evasion Rating while on Full Life (50-70)% increased Evasion Rating 30% increased Movement Speed when on Full Life Damage of Enemies Hitting you is Unlucky while you are on Full 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 of Enemies Hitting you is Lucky Nearby Allies' Damage with Hits is LuckyHe was a lot luckier before he lost this.	Amulet	29	(12-20)% increased Rarity of Items found+(40-50) to Dexterity (5-8)% increased Movement Speed Lightning Damage of 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 of 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.	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 of Enemies Hitting you is Unlucky while you are on Low Life
Relic of the PactRelic of the Pact Sage WandWand Physical Damage: 23-42 Critical Strike Chance: 8.00% Attacks per Second: 1.20Requires Level 30, 119 Int(17-21)% 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.	Wand	30	(17-21)% increased Spell DamageGrants Level 1 Blood Sacrament Skill Your Critical Strike Chance is Lucky while on Low Life
HyperboreusHyperboreus Leather BeltRequires Level 53+(25-40) to maximum Life+(30-40)% to Cold Resistance Chill nearby Enemies when you Focus, causing 30% reduced Action Speed Focus has (30-50)% increased Cooldown Recovery Rate (50-70)% increased Damage with Hits and Ailments against Chilled Enemies Veiled SuffixCold winds whirl at the crown of the world.Take this item to Jun Ortoi to have her Unveil it.	Belt	53	+(25-40) to maximum Life+(30-40)% to Cold Resistance Chill nearby Enemies when you Focus, causing 30% reduced Action Speed Focus has (30-50)% increased Cooldown Recovery Rate (50-70)% increased Damage with Hits and Ailments against Chilled Enemies Veiled Suffix +(16-20)% to Cold and Chaos Resistances +(31-35) to Dexterity and Intelligence Regenerate 200 Energy Shield per second while a Rare or Unique Enemy is Nearby +(16-20)% to Fire and Chaos Resistances +(16-20)% to Lightning and Chaos Resistances Your Critical Strike Chance is Lucky while Focused Focus has (5-8)% increased Cooldown Recovery Rate +(31-35) to Strength and Dexterity +(31-35) to Strength and Intelligence (14-16)% increased Trap Throwing Speed
The FulcrumThe Fulcrum Ezomyte StaffWarstaff Physical Damage: (127.2-148.4) to (384-448) Critical Strike Chance: 7.30% Attacks per Second: 1.25 Weapon Range: 1.3 metresRequires 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.	Warstaff	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
Triumvirate AuthorityTriumvirate Authority Unset RingRequires Level 64Has 1 Socket+2 to Level of Socketed Vaal Gems <Three random Socketed Vaal Skill modifiers>The priest that carried Yaomac's sigil was granted one boon by each serpentine head.	Ring	64	Has 1 Socket+2 to Level of Socketed Vaal Gems <Three random Socketed Vaal Skill modifiers> Socketed Vaal Skills have 60% increased Area of Effect Socketed Vaal Skills have 50% increased Aura Effect Socketed Vaal Skills deal 150% more Damage Socketed Vaal Skills grant Elusive when Used Hits from Socketed Vaal Skills ignore Enemy Monster Physical Damage Reduction Hits from Socketed Vaal Skills ignore Enemy Monster Resistances Damage with Hits from Socketed Vaal Skills is Lucky Socketed Vaal Skills have 80% increased Projectile Speed Socketed Vaal Skills have 80% increased Skill Effect Duration Socketed Vaal Skills have 30% reduced Soul Gain Prevention Duration Socketed Vaal Skills require 30% less Souls per Use You have Tailwind if you've used a Socketed Vaal Skill Recently Socketed Vaal Skills have 20% chance to regain consumed Souls when used
Triumvirate AuthorityTriumvirate Authority Unset RingRequires Level 64Has 1 Socket+2 to Level of Socketed Vaal Gems <Four random Socketed Vaal Skill modifiers>The priest that carried Yaomac's sigil was granted one boon by each serpentine head.	Ring	64	Has 1 Socket+2 to Level of Socketed Vaal Gems <Four random Socketed Vaal Skill modifiers> Socketed Vaal Skills have 60% increased Area of Effect Socketed Vaal Skills have 50% increased Aura Effect Socketed Vaal Skills deal 150% more Damage Socketed Vaal Skills grant Elusive when Used Hits from Socketed Vaal Skills ignore Enemy Monster Physical Damage Reduction Hits from Socketed Vaal Skills ignore Enemy Monster Resistances Damage with Hits from Socketed Vaal Skills is Lucky Socketed Vaal Skills have 80% increased Projectile Speed Socketed Vaal Skills have 80% increased Skill Effect Duration Socketed Vaal Skills have 30% reduced Soul Gain Prevention Duration Socketed Vaal Skills require 30% less Souls per Use You have Tailwind if you've used a Socketed Vaal Skill Recently Socketed Vaal Skills have 20% chance to regain consumed Souls when used
Viridi's VeilViridi's Veil Praetor CrownArmour: (354-481) Energy Shield: (112-150)Requires 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	Helmet	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
Voice of the StormVoice of the Storm Lapis AmuletRequires Level 69+(20-30) to Intelligence50% increased Lightning Damage +(10-15) to all Attributes (10-20)% increased maximum Mana Lightning Damage with Non-Critical Strikes is LuckyWhen she of many mouths spoke, we bowed in awe and were crushed. This item can be transformed with a Blessing of Esh	Amulet	69	+(20-30) to Intelligence50% increased Lightning Damage +(10-15) to all Attributes (10-20)% increased maximum Mana Lightning Damage with Non-Critical Strikes is Lucky