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 effective lucky critical strike chance can be calculated from the following formula: In the above formula, we find the percentage that we do NOT crit, multiply it by our crit chance, then add this to our original crit chance.
Example problem: The lucky critical strike chance for our default critical strike chance is 40%. It would be calculated as follows:
It should be noted that these calculations neglect the effect of accuracy on critical strikes.
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.)
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.
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:
The total probability to get a lucky critical strike is the sum of 1 and 2.
Related items
Base items
Item | Item Class | Buff Effects | |
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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
Item | Item Class | Stats | |||
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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
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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
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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
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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 |
References
- ↑ Mark_GGG (May 13, 2013 4:01 AM). "p/c new ring lori's lantern". Path of Exile Forum. Retrieved September 22, 2013.