Luck



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 refer specifically to damage ranges, and not related mechanics such as accuracy or critical strike chance.

Lucky and Unlucky modifiers affecting the same thing will cancel each other out.

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.

Binary Rolls


More specifically, the lucky critical strike chance can be calculated from the following formula: $CritChance_{Lucky} = 2 \times CritChance_{Default} - {CritChance_{Default} } ^ 2$

For example, the lucky critical strike chance for an default critical strike chance of 40% could be calculated as follows: $CritChance_{Lucky} = 2 \times 40%- 40% ^ 2 = %$

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

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: $B = 1 - A$

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 $$ multiplied with A: $C = B \times A$

The total probability to get a lucky critical strike is the sum of A and $$. $\begin{align} CritChance_{Lucky} & = A + C \\ & = A + B \times A \\ & = A + (1 - A) \times A \\ & = 2 \times A - A^2 \end{align}$

Damage in a Range
Given an integer roll between $min$ and $max$, the normal expected value is $\frac{min+max}{2}$.

A lucky roll has expected value $\frac{min}{3} + \frac{2*max}{3} + \frac{max-min}{6*(1+max-min)}$.

Note that the last term in this is always between $0$ and $1/6$ 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 $\frac{min+2*max}{3}$.

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. 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. 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 $0$ then we get a $33%$ more multiplier. As the minimum damage increases our percentage increase decreases.

Base items
The following base items are related to Luck:

Unique items
The following unique items are related to Luck:

Related skill gems
The following skill gems are related to Luck:

Passive skills
The following passive skills are related to Luck:

Keystone passive skills
The following keystone passive skills are related to Luck:

Related modifiers
The following modifiers are related to Luck: