Luck
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.[1]
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
More specifically, the lucky critical strike chance can be calculated from the following formula:
For example, the lucky critical strike chance for an default critical strike chance of 40% could be calculated as follows:
It should be noted that these calculations neglect the effect of accuracy on critical strikes.
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:
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:
The total probability to get a lucky critical strike is the sum of A and 2.
Damage in a Range
Given an integer roll between and , the normal expected value is .
A lucky roll has expected value .
Note that the last term in this is always between and 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 .
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: 370%
Projectile Speed: 1040
AoE Radius: 9Unleashes a ball of fire towards a target which explodes, damaging nearby foes.Deals (9-1640) to (14-2460) Fire Damage
25% chance to Ignite enemies
Base radius is (0.9-1.4) metres
Additional Effects From 1-20% Quality:
Base radius is (0.01-0.2) metresPlace 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: 190%
Projectile Speed: 420Launches unpredictable sparks that move randomly until they hit an enemy or expire.Deals (1-104) to (28-1983) Lightning Damage
Base duration is 2.00 seconds
Fires (4-8) Projectiles
Additional Effects From 1-20% Quality:
Fires +(0.1-2) ProjectilesPlace 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 then we get a more multiplier. As the minimum damage increases our percentage increase decreases.
Related items
Base items
The following base items are related to Luck:
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 |
Ritual Scarab of SelectivenessRitual Scarab of SelectivenessStack Size: 20Rerolling Favours at Ritual Altars in Area has no cost the first 2 times Ritual Altars in Area allow rerolling Favours 1 additional timePick and choose which sacred tenets you prefer...Can be used in a personal Map Device to add modifiers to a Map. | Map Fragment | 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
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+(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 | Chain 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. | Wild Leather | 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. | Gold 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. | 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 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. | Sage Wand | 30 | (17-21)% increased Spell DamageGrants Level 1 Blood Sacrament Skill Your Critical Strike Chance is Lucky while on Low Life | ||
Bound FateBound Fate Cloth BeltRequires Level 49(15-25)% increased Stun and Block Recovery+(20-30) to Dexterity +(20-30) to Intelligence +(60-80) to maximum Life Every 5 seconds, gain one of the following for 5 seconds: Your Hits are always Critical Strikes Hits against you are always Critical Strikes Attacks cannot Hit you Attacks against you always Hit Your Damage with Hits is Lucky Damage of Hits against you is LuckyKnowing what will happen and knowing when it will happen are two very different things. | Cloth Belt | 49 | (15-25)% increased Stun and Block Recovery+(20-30) to Dexterity +(20-30) to Intelligence +(60-80) to maximum Life Every 5 seconds, gain one of the following for 5 seconds: Your Hits are always Critical Strikes Hits against you are always Critical Strikes Attacks cannot Hit you Attacks against you always Hit Your Damage with Hits is Lucky Damage of Hits against you is Lucky | ||
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. | Leather 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
| ||
The Iron FortressThe Iron Fortress Crusader PlateArmour: (1081-1382) Movement Speed: -5%Requires Level 59, 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 | 59 | +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 | ||
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. | 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 | ||
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. | Unset Ring | 64 | Has 1 Socket+2 to Level of Socketed Vaal Gems
| ||
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. | Unset Ring | 64 | Has 1 Socket+2 to Level of Socketed Vaal Gems
| ||
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 | 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 | ||
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 | Lapis 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 |
Related skill gems
The following skill gems are related to Luck:
Gem | ||||
---|---|---|---|---|
Vaal GraceVaal GraceAura, Spell, AoE, Duration, Vaal Level: (1-20) Cooldown Time: 0.50 sec Can Store 1 Use(s) Souls Per Use: 50 Can Store 1 Use(s) Soul Gain Prevention: 9 sec Cast Time: Instant AoE Radius: 36Requires Level 24Casts a temporary aura that grants you and your allies additional chance to evade attacks, and makes suppressed spell damage unlucky against you.Base duration is 6.00 seconds Modifiers to Buff Duration also apply to this Skill's Soul Gain Prevention +(0-1.9) metres to radius You and nearby allies have +15% chance to Evade Attack Hits Suppressed Spell Damage dealt to you is Unlucky Additional Effects From 1-20% Quality: (2-40)% increased Area of EffectCorruptedPlace into an item socket of the right colour to gain this skill. Right click to remove from a socket. | 24 | N/A | N/A | |
Vaal ArcVaal ArcSpell, Chaining, Vaal, Lightning, Duration Level: (1-20) Souls Per Use: 25 Can Store 1 Use(s) Soul Gain Prevention: 2 sec Cast Time: 0.70 sec Critical Strike Chance: 6.00% Effectiveness of Added Damage: 220%Requires Level 12A shocking arc of lightning reaches from the caster to a targeted enemy and chains to other enemies. Each time the arc chains from an enemy, it will do so simultaneously to two further enemies, but never to enemies that have already been hit by any of these chains. Also grants a buff making your Arc damage lucky for a short duration.Deals (38-924) to (64-1539) Lightning Damage 100% increased Duration of Lightning Ailments Base duration is 4.00 seconds 100% chance to Shock enemies Chains +(5-9) Times 15% more Damage with Hits and Ailments for each remaining Chain 100% increased Effect of Lightning Ailments Additional Effects From 1-20% Quality: Chains +(0.05-1) TimesCorruptedPlace into an item socket of the right colour to gain this skill. Right click to remove from a socket. | 12 | N/A | N/A |
Related passive skills
Passive skills
The following passive skills are related to Luck:
Name | Stats |
---|---|
Blight Chest Chance and Lucky Chests | Varieties of Items contained in 1 Blight Chest in your Maps are Lucky Lanes of Blight Encounters in your Maps have 5% chance for an additional Reward Chest [1] |
Lucky Blight Chests | 8% chance for Blight Chests to contain an additional Reward Varieties of Items contained in 1 Blight Chest in your Maps are Lucky [1] [2] |
Epidemiology | Blight Chests in your Maps have 80% more chance to contain Blighted Maps Varieties of Items contained in 3 Blight Chests in your Maps are Lucky [1] |
Keystone passive skills
The following keystone passive skills are related to Luck:
Name | Stats |
---|---|
Dance with Death |
Can't use Helmets |
Related modifiers
The following modifiers are related to Luck:
Name | Generation Type | Stats | Spawn Weighting | |
---|---|---|---|---|
of the Order | Suffix | 72 | Your Critical Strike Chance is Lucky while Focused | belt 0 |
of the Order | Suffix | 50 | Your Critical Strike Chance is Lucky while Focused | belt 0 |
of the Order | Suffix | 60 | Your Critical Strike Chance is Lucky while Focused Focus has (5-8)% increased Cooldown Recovery Rate | belt 1000 |
HellscapeDownsideDamageIsUnlucky1___ | Scourge Detriment | 45 | Damage with Hits is Unlucky | amulet 500 |
References
- ↑ Mark_GGG (May 13, 2013 4:01 AM). "p/c new ring lori's lantern". Path of Exile Forum. Retrieved September 22, 2013.