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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

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:

 \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 (1) multiplied with A:

 \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 A 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)

Damage in a Range

Given an integer roll between  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} min
    and  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} max
    , the normal expected value is  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} \frac{min+max}{2}
    .

A lucky roll has expected value  \pagecolor[RGB]{15,15,15} \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  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} 0
    and  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} 1/6
    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  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} \frac{min+2*max}{3}
    .

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
Radius: 9
Mana Cost: (6-25)
Cast Time: 0.75 sec
Critical Strike Chance: 6.00%
Effectiveness of Added Damage: 180%
Projectile Speed: 1040
Unleashes a ball of fire towards a target which explodes, damaging nearby foes.Per 1% Quality:
0.5% chance to Ignite enemies
Deals (9-1095) to (14-1643) Fire Damage
(20-39)% chance to Ignite enemies
(50-88)% more Damage with Ignite
+(0-4) to radius
Place into an item socket of the right colour to gain this skill. Right click to remove from a socket.
Fireball skill icon.png
. 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
Mana Cost: (5-21)
Cast Time: 0.65 sec
Critical Strike Chance: 6.00%
Projectile Speed: 560
Launches unpredictable sparks that move randomly until they hit an enemy or expire.Per 1% Quality:
1% increased Projectile Speed
Deals (1-63) to (20-1198) Lightning Damage
Base duration is 2 seconds
Fires (4-6) additional Projectiles
(0-19)% increased Projectile Speed
Place into an item socket of the right colour to gain this skill. Right click to remove from a socket.
Spark skill icon.png
. 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  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} 0
    then we get a  \pagecolor[RGB]{15,15,15} \color[RGB]{163,141,109} 33%
    more multiplier. As the minimum damage increases our percentage increase decreases.

Related items

Base items

The following base items are related to Luck:

ItemItem ClassLvl.Buff Effects
Diamond FlaskDiamond 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

The following unique items are related to Luck:

ItemBase ItemLvl.Stats
Coward's ChainsCoward's Chains
Chain Belt
Requires Level 22+(9-20) to maximum Energy Shield+(10-15) to all Attributes
(20-25)% increased Damage
(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
Chain Belt22+(9-20) to maximum Energy Shield+(10-15) to all Attributes
(20-25)% increased Damage
(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 ToePerquil'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.
Gold Amulet29(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 LanternLori'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.
Prismatic Ring30+(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 FortuneFox's Fortune
Wild Leather
Quality: +20%
Evasion: (1572-2038)
Movement Speed: -3%
Requires Level 54, 73 Dex+(20-30) to Dexterity
Adds 5 to 12 Physical Damage to Attacks
+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.
Wild Leather54+(20-30) to Dexterity
Adds 5 to 12 Physical Damage to Attacks
+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
The Iron FortressThe Iron Fortress
Crusader Plate
Quality: +20%
Armour: (1202-1322)
Movement Speed: -5%
Requires Level 68, 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 Plate68+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

Related skill gems

The following skill gems are related to Luck:

Skill gemStr.Dex.Int.Lvl.
Vaal ArcVaal ArcVaal, Spell, Chaining, Lightning, Duration
Souls Per Use: 25 (N) /
Can Store 1 Use(s)
Soul Gain Prevention: 6.000 sec
Cast Time: 0.80 sec
Critical Strike Chance: 5.00%
Effectiveness of Added Damage: 180%
Requires Level 12A shocking arc of lightning stretches from the caster to a targeted enemy and chains to other nearby enemies. Each time the beam chains it will also chain simultaneously to a second enemy, but no enemy can be hit twice by the beams. Also grants a buff making you lucky when damaging enemies with Arc for a short duration.Per 1% Quality:
1.5% increased Shock Duration on enemies
Base duration is 4 seconds
Deals (38-924) to (64-1539) Lightning Damage
100% chance to Shock enemies
Chains +(5-9) Times
100% increased Effect of Shock
100% increased Shock Duration on enemies
15% more Damage for each remaining Chain
Modifiers to Buff Duration also apply to this Skill's Soul Gain Prevention
Corrupted
Place into an item socket of the right colour to gain this skill. Right click to remove from a socket.
Vaal Arc skill icon.png
N/AN/Ayes12

Related passive skills

Passive skills

The following passive skills are related to Luck:

NameStats
Supercharge
IncreasedLightningDamage passive skill icon.png
Lightning Damage with Non-Critical Strikes is Lucky [1]

Keystone passive skills

The following keystone passive skills are related to Luck:

Name Stats
Dance with Death
KeystoneDancewithDeath passive skill icon.png

Can't use Helmets
Your Critical Strike Chance is Lucky
Your Damage with Critical Strikes is Lucky
Enemies' Damage with Critical Strikes against you is Lucky

Related modifiers

The following modifiers are related to Luck:

NameGeneration TypeFor generated item/monster modifiers the minimum item/monster level respectively. Some generation types may not require this condition to be met, however item level restrictions may be raised to 80% of this value.StatsSpawn Weighting
of the OrderSuffix50Your Critical Strike Chance is Lucky while Focussedbelt 0
default 0
of the OrderSuffix72Your Critical Strike Chance is Lucky while Focussedbelt 2000
default 0

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.