Luck

Some rolls in the game are Lucky or Unlucky. Lucky means that a number is rolled twice and the best result is picked. 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 damage roll is unlucky, the damage calculation will be rolled twice and the worst result will be picked instead. Rolls to critical strike chance/evasion/dodge/block are unchanged and only happen once.

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: $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}CritChance_{Lucky}=2\times CritChance_{Default}-{CritChance_{Default}}^{2}$

For example, the lucky critical strike chance for an default critical strike chance of 40% could be calculated as follows: $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}CritChance_{Lucky}=2\times 40\%-40\%^{2}=64\%$

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

Derivation of lucky rolls ▾ [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: $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}B=1-A$

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: $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}C=B\times A$

The total probability to get a lucky critical strike is the sum of 1 and 2. $\color [rgb]{0.6392156862745098,0.5529411764705883,0.42745098039215684}{\begin{aligned}CritChance_{Lucky}&=A+C\\&=A+B\times A\\&=A+(1-A)\times A\\&=2\times A-A^{2}\end{aligned}}$