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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]}
Contents
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.
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:

(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:

(2)
The total probability to get a lucky critical strike is the sum of (1) and (2).

(3)
Related items
Base items
Item  Item Class  Buff Effects  

Diamond FlaskDiamond FlaskLasts 4.00 Seconds Consumes 20 of 40 Charges on use Your Critical Strike Chance is LuckyRequires Level 27Right click to drink. Can only hold charges while in belt. Refills as you kill monsters.  Critical Utility Flasks  27  Your Critical Strike Chance is Lucky 
Unique items
Item  Item Class  Stats  

Coward's ChainsCoward's Chains Chain BeltRequires Level 22+(920) to maximum Energy Shield(2025)% increased Damage +(1015) to all Attributes (510)% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are Cursed with Vulnerability You are Cursed with Level 10 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  Belts  22  +(920) to maximum Energy Shield(2025)% increased Damage +(1015) to all Attributes (510)% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are Cursed with Vulnerability You are Cursed with Level 10 Vulnerability 
Perquil's ToePerquil's Toe Gold AmuletRequires Level 29(1220)% increased Rarity of Items found+(4050) to Dexterity (58)% 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.  Amulets  29  (1220)% increased Rarity of Items found+(4050) to Dexterity (58)% 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+(810)% to all Elemental Resistances+10% to all Elemental Resistances (68)% increased Movement Speed when on Low Life 31% increased Light Radius +(2025)% 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.  Rings  30  +(810)% to all Elemental Resistances+10% to all Elemental Resistances (68)% increased Movement Speed when on Low Life 31% increased Light Radius +(2025)% 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: (15722038) Movement Speed: 3%Requires Level 54, 73 DexAdds 5 to 12 Physical Damage to Attacks +(2030) to Dexterity +1500 to Evasion Rating while on Full Life +(400500) to Evasion Rating (120150)% 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.  Body Armours  54  Adds 5 to 12 Physical Damage to Attacks +(2030) to Dexterity +1500 to Evasion Rating while on Full Life +(400500) to Evasion Rating (120150)% increased Evasion Rating 10% increased Movement Speed Damage from Enemies Hitting you is Unlucky while you are on Full Life 
References
 ↑ Mark_GGG (May 13, 2013 4:01 AM). "p/c new ring lori's lantern". Path of Exile Forum. Retrieved September 22, 2013.