Luck

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.

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:

$\color[RGB]{163,141,109} CritChance_{Lucky} = 2 \times CritChance_{Default} - {CritChance_{Default} } ^ 2$

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

$\color[RGB]{163,141,109} CritChance_{Lucky} = 2 \times 40$

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

Damage in a Range

Given an integer roll between $\color[RGB]{163,141,109} min$ and $\color[RGB]{163,141,109} max$, the normal expected value is $\color[RGB]{163,141,109} \frac{min+max}{2}$. A lucky roll has expected value $\color[RGB]{163,141,109} \frac{min}{3} + \frac{2*max}{3} + \frac{max-min}{6*(1+max-min)}$. Note that the last term in this is always between $\color[RGB]{163,141,109} 0$ and $\color[RGB]{163,141,109} 1/6$ and therefore is negligible (it is a byproduct of the fact our uniform variable only takes values over the integers.)

How much does lucky improve our expected value - if our minimum damage is $\color[RGB]{163,141,109} 0$ then we get a $\color[RGB]{163,141,109} 33$ more multiplier. As the minimum damage increases our percentage increase decreases.

Derivation of lucky rolls ▾ [Expand] ▾

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:

$\color[RGB]{163,141,109} B = 1 - A$

(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:

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

(2)

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

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

(3)

Related items

Base items

Item	Item Class		Buff Effects
Diamond FlaskDiamond FlaskLasts 4.00 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 Flasks	27	100% increased Global Critical Strike Chance

Unique items

Item	Item Class		Stats
Coward's ChainsCoward's Chains Chain BeltRequires Level 22+(9-20) to maximum Energy Shield+(10-15) to all Attributes (5-10)% increased Movement Speed Damage from 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, with 40% increased EffectAs 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	+(9-20) to maximum Energy Shield+(10-15) to all Attributes (5-10)% increased Movement Speed Damage from 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, with 40% increased Effect
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 from Enemies Hitting you is Lucky Nearby Allies' Damage with Hits is LuckyHe was a lot luckier before he lost this.	Amulets	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
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 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	+(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 LeatherQuality: +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 LifeFollow your most basic instincts. The rest will come naturally.	Body Armours	54	+(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 FulcrumThe Fulcrum Ezomyte StaffWarstaff Quality: +20% Physical Damage: (137.8-159) to (416-480) Critical Strike Chance: 7.30% Attacks per Second: 1.25 Weapon Range: 13Requires 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.	Warstaves	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