Luck

Some rolls in the game are Lucky or Unlucky. Lucky rolls will be made twice, and the higher value is chosen. Diamond flasksDiamond FlaskLasts 6.00 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. makes the critical strike chance lucky for a duration.

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.

Unlucky

If a roll is unlucky, it will be rolled twice and the lower value will be chosen. Currently unlucky is only found on the unique ring 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 of 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..

Effectiveness

On percentage rolls, such as critical chance, Lucky/unlucky rolls will be up to a 100% more/less modifier, changing linearly towards 0% as critical chance increases. On 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

Lucky crit rolls visualized in a tree diagram.

More specifically, the lucky critical strike chance can be calculated from the following formula:

LuckyCritChance = 2*DefaultCritChance - DefaultCritChance^2

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

LuckyCritChance = 2*40% - 40%*40% = 80% - 1600%% = 80% - 16% = 64%

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

Derivation of lucky rolls

This is a problem 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: B = 1 - A 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: C = B * A The total probability to get a lucky critical strike is the sum of A and C. LuckyCritChance = A + C = A + B * A = A + (1 - A) * A = 2*A - A^2