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Luck

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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.[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

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:

 \pagecolor[RGB]{15,15,15} \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:

 \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} CritChance_{Lucky} = 2 \times 40%- 40% ^ 2 = 64%

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

Derivation of lucky rolls

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:

 \pagecolor[RGB]{15,15,15} \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:

 \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} C = B \times A

 

 

 

 

(2)

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

 \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} \begin{align}
CritChance_{Lucky} & = A + C \\
& = A + B \times A \\
& = A + (1 - A) \times A \\
& = 2 \times A - A^2
\end{align}

 

 

 

 

(3)

Related items

Base items

ItemItem ClassLvl.Buff Effects
Diamond Flask
Diamond FlaskLasts 4.00 Seconds
Consumes 20 of 40 Charges on use
Your Critical Strike Chance is Lucky
Requires Level 27Right click to drink. Can only hold charges while in belt. Refills as you kill monsters.
Critical Utility Flasks27Your Critical Strike Chance is Lucky

Unique items

ItemItem ClassLvl.Stats
Coward's Chains
Coward's Chains
Chain Belt
Requires Level 22+(9-20) to maximum Energy Shield(20-25)% increased Damage
+(10-15) to all Attributes
(5-10)% increased Movement Speed
Damage from Enemies Hitting you is Unlucky while you are Cursed with Vulnerability
You are Cursed with Level 10 Vulnerability
As 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
Belts22+(9-20) to maximum Energy Shield(20-25)% increased Damage
+(10-15) to all Attributes
(5-10)% 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 Toe
Perquil's Toe
Gold Amulet
Requires 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 Lucky
He was a lot luckier before he lost this.
Amulets29(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 Lantern
Lori's Lantern
Prismatic Ring
Requires 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 Life
By 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.
Rings30+(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 Fortune
Fox's Fortune
Wild Leather
Quality: +20%
Evasion: (1572-2038)
Requires Level 54, 73 Dex3% reduced Movement Speed (Hidden)Adds 5 to 12 Physical Damage to Attacks
+(20-30) to Dexterity
+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
Follow your most basic instincts.
The rest will come naturally.
Body Armours543% reduced Movement Speed (Hidden)Adds 5 to 12 Physical Damage to Attacks
+(20-30) to Dexterity
+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

References

  1. Mark_GGG (May 13, 2013 4:01 AM). "p/c new ring lori's lantern". Path of Exile Forum. Retrieved September 22, 2013.