# Lucky

For the equipment, see Lucky equipment.
 This article has a calculator here.Calculators determine experience and costs based on real-time prices from the Grand Exchange Market Watch.

Lucky is an Invention perk which has a 0.5% chance per rank when hit that the damage received will be reduced to 1. It can be created in armour gizmos.

This perk does not stack with Saradomin or Zamorak warpriest armour. Lucky works on all damage types, including soft typeless (excluding hard typeless) damage taken in combat. This means that it works on reflected damage and Helwyr's bleeds, but does not activate for Araxxor's cleave or when trying to eat a dwarven rock cake (cooled).[source needed] Additionally, the lucky perk does not work in PvP situations.

When the perk triggers, it prompts the following message: Your Lucky perk reduces the damage of this attack by [number].

Average damage reduction for the different ranks of the Lucky perk.[r 1]
Rank Activation chance Average damage reduction
1 0.5% (0.55%) 0.5% (0.55%)
2 1.0% (1.10%) 1.0% (1.10%)
3 1.5% (1.65%) 1.5% (1.65%)
4 2.0% (2.20%) 2.0% (2.20%)
5 2.5% (2.75%) 2.5% (2.75%)
6 3.0% (3.30%) 3.0% (3.30%)
1. ^ All numbers in parentheses refer to gear with item level 20.
Calculations
Notes
• Assumption :
• All hits that are taken have the possibility to proc Lucky.
• ${\displaystyle R}$ is the rank of the Lucky perk.
• ${\displaystyle p_{R}}$ is the proc chance of Lucky to activate (if the perk is on level 20 gear, then this value is multiplied by 1.1).
• ${\displaystyle \textstyle \sum _{n=0}^{\infty }\left(1-p\right)^{n}p=1}$ when ${\displaystyle 0\leq p<1}$ or ${\displaystyle \left|1-p\right|<1}$. This represents summing over all possibilities where integer ${\displaystyle n\in [0,\infty )}$ represents the amount of hits prior to the proc of Lucky. The probability of Lucky proccing on hit ${\displaystyle {n+1}}$ is therefore ${\displaystyle \left(1-p\right)^{n}p}$.
• ${\displaystyle d_{n,i}^{(j)}}$ is the ${\displaystyle i^{th}}$ random value uniformly sampled between the enemy's minimum hit and maximum hit. The ${\displaystyle j}$ describes different sets of hits. The sets are regenerated for every new value of ${\displaystyle n}$.
• ${\displaystyle d_{n,i}^{(0)}}$ has ${\displaystyle n}$ elements with integer ${\displaystyle i\in [1,n]}$. This is the damage taken before the perk procs.
• Clarification : This does mean that for ${\displaystyle n=0}$ that there are no elements in this set.
• ${\displaystyle d_{n}^{(1)}}$ is the damage taken from a single hit if the Lucky perk were to proc (without having it on gear).
• The set of values in ${\displaystyle d_{n}^{(j)}}$ for integer ${\displaystyle j\in [0,1]}$ is the same in both the numerator and denominator of the ratios for any given ${\displaystyle n}$.

Damage reduction
• The average ratio of damage taken, ${\displaystyle r_{avg}}$, from Lucky to that of without Lucky is
${\displaystyle r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+1\right]}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+d_{n}^{(1)}\right]}}}$
• The average damage reduction is then, after some rearranging,
${\displaystyle 1-r_{avg}={\frac {\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left(d_{n}^{(1)}-1\right)}{\sum \limits _{n=0}^{\infty }\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum \limits _{i=1}^{n}d_{n,i}^{(0)}\right)+d_{n}^{(1)}\right]}}}$
Simplifications
• This can be simplified if the assumption is that every value ${\displaystyle d_{n,i}^{(0)}}$ ${\displaystyle \forall }$ integer ${\displaystyle i\in [1,n]}$, ${\displaystyle d_{n}^{(1)}}$ ${\displaystyle \forall }$ integer ${\displaystyle n\in [0,\infty )}$ is taken to be the same ${\displaystyle \left(d_{n,i}^{(0)}=d_{n}^{(1)}\rightarrow d\right)}$. In this scenario, there is no random element and the above ${\displaystyle r_{avg}}$ is then only dependent on the proc chance. A table is provided at the top of the page.

Using these simplifications, the damage reduction reduces to

${\displaystyle 1-r_{avg}=p_{R}}$

## Sources

MaterialRarityPerk ranks with X materials
Standard gizmoAncient gizmo
12345123456789
Silent componentsRare1234–5501–22345666
Delicate partsCommon0011–21–2000111–21–21–21–3
Light componentsUncommon01–21–21–31–5011–21–21–32–42–52–62–6
Fortunate componentsRare11–21–31–51–501–21–21–32–32–32–32–32–3

### Suggested gizmos

Gizmo layout Possible perks
• Other possible perks:
• Other possible perks:
• Other possible perks:
• Other possible perks:
• Other possible perks:

## Update history

The update history project is a work-in-progress – not all updates to this topic may be covered below. See here for how to help out!
• ninja 18 July 2016 (Update):
• The Lucky perk no longer triggers while the Resonance ability is in effect.
• patch 15 February 2016 (Update):
• The Lucky Perk message is now filterable.