View source for Luck

[[File:LuckyCritRolls.png|thumb|right|upright=2|Lucky crit rolls visualized in a [[wikipedia:Tree_diagram_(probability_theory)|tree diagram]].]]

'''Luck''' is a mechanic that rolls an outcome or a number in a range twice, with '''Lucky''' rolls applying the ''best'' result, and '''Unlucky''' rolls applying the ''worst'' result. 

Modifiers that specify {{c|mod|Damage}} or {{c|mod|when Damaging}} and luck refer specifically to damage ranges after any increased/more modifiers (or "tooltip damage range"), not to the base damage of the weapon/skill. These stats do not affect other mechanics like chance to hit/[[accuracy]] or [[critical hit]] chance.<ref name="Lori"/> Additionally, these stats only apply to hit damage, and have no effect on the damage roll of [[damage over time]] from damaging [[ailment]]s, which roll entirely independently of the hit damage's roll.<ref>Confirmed through direct testing by [[Prohibited Library (community)]]. https://discord.com/channels/991073626721763429/991092346651287623/1165229624595316807</ref>

Lucky and Unlucky modifiers affecting the same stat or mechanic will cancel each other out.<ref>https://www.pathofexile.com/forum/view-thread/3108809/page/1#p24083466</ref> 

==Binary Rolls==
[[File:Lucky.PNG|thumb|right|upright=2|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.]]

A binary roll applies to something with only two choices such as chance of hitting, critical strikes, or chance of blocking. The Lucky or Unlucky chance can be calculated from your original chance with the following formulas:

{{Math|formula = Lucky = 2 \times Chance - {Chance}^2 }}

{{Math|formula = Unlucky = {Chance}^2 }}

For example, the lucky critical strike chance for a critical strike chance of 40% could be calculated as follows:

{{Math|formula = Lucky = 2 \times 0.4- 0.4 ^ 2 = 0.8 - 0.16 = 0.64}}

So a crit chance of 40% will give a lucky crit chance of 64%.

{| class="wikitable"
|+ Lucky/unlucky effect on binary rolls
|-
! Original chance!! Lucky chance !! Unlucky chance
|-
| 10% || 19% || 1%
|-
| 25%|| 43.75% || 6.25%
|-
| 50% || 75% || 25%
|-
| 75%|| 93.75% || 56.25%
|-
| 90% || 99% || 81%
|}

Note that if a roll has 0% or 100% chance, lucky and unlucky will have no effect.

==Damage in a Range==
===Lucky Damage===
Without luck, the average or expected value for a damage roll is {{math|formula=\frac{min+max}{2} }}. 

A lucky roll has expected value {{math|formula= \frac{min + 2 \times max}{3} }}. 

The extra damage luck adds is a sixth of the range, or {{math|formula= \frac{max - min}{6} }}. 

The average damage is effectively nudged closer toward the max damage, and further from the minimum damage. This means that the larger the range, the more effect luck will have.

Example 1:
 You use an attack with a damage range of 600-800, for an average of {{math|formula= \frac{600+800}{2} = 700 }}.
 The range, or difference between minimum and maximum, is {{math|formula= 800 - 600 = 200 }}.
 The bonus damage will be {{math|formula= \frac{200}{6} = 33.3 }} so your new average damage is {{math|formula=700+33.3 = 733.3 }}
 To calculate how much more damage this is, {{math|formula= \frac{733.33}{700} = 1.047}} which means a 4.7% increase.

You can calculate how much more damage lucky will provide as a percentage directly as {{math|formula= \frac{range}{6 \times average} \times 100\% }}.

Example 2: 
 You use a spell with a damage range of 100-1000, for an average of {{math|formula= \frac{100+1000}{2} = 550 }} and range of {{math|formula= 1000-100=900 }}.
 Adding lucky will give {{math|formula= \frac{900}{6 \times 550} = 0.273 \times 100\% = 27.3\%}} more damage.

If your minimum damage is 0 this gives a 33% more multiplier. As the minimum damage increases this scales down to a 0% more multiplier when the minimum damage equals the maximum.

===Unlucky Damage===
Unlucky damage nudges the damage the opposite way, giving an expected value {{math|formula= \frac{2 \times min + max}{3} }}, which has subtracted a sixth of the range from the original average.

Making damage unlucky will give a less damage multiplier equal to the more damage multiplier that lucky would give, so the same formula as above can be used.

==Related items==
===Unique items===
The following [[unique item]]s are related to Luck:
{{Drop enabled unique item table
|tables=item_stats
|join=items._pageID=item_stats._pageID
|where = item_stats.id LIKE "%rolls%"
      OR items.stat_text LIKE "%Lucky%"
|large=1
|base_item=1
|level=1
|stat=1
}}


==Related passive skills==
===Passive skills===
The following [[passive skill]]s are related to Luck:
{{Query base passive skills
|where=passive_skills.stat_text LIKE "%luck%"
}}

===Ascendancy passive skills===
The following [[Ascendancy passive skill]]s are related to Luck:
{{Query ascendancy passive skills
|where=passive_skills.stat_text LIKE "%luck%"
}}

===Keystone passive skills===
The following [[keystone passive skill]]s are related to Luck:
{{Passive skill table
|q_where=passive_skills.stat_text LIKE "%luck%"
     AND passive_skills.is_keystone = 1
|large=1
|stat=1
}}


==Related modifiers==
The following [[modifier]]s are related to Luck:
{{modifier table
|q_tables=mod_stats
|q_join=mods._pageID=mod_stats._pageID
|q_where = mod_stats.id LIKE "%rolls%" 
       AND mod_stats.id NOT LIKE "%scrolls%"
       AND mods.domain NOT IN ("25")
       AND mods.id NOT LIKE "%Unique%"
       AND mods.id NOT LIKE "%Ritual%"
|generation_type=1
|level=1
|stat_text=1
|spawn_weights=1
}}

==References==
{{reflist|refs=
<ref name="Lori">{{cite web|author=Mark_GGG|date=May 13, 2013 4:01 AM|title=p/c new ring lori's lantern|url=https://www.pathofexile.com/forum/view-thread/377182/page/3#p3349323|publisher=Path of Exile Forum|accessdate= September 22, 2013}}</ref>
}}

{{sister|poewiki}}

[[Category:Game mechanics]]