Lucky

For the equipment, see lucky equipment

Lucky
Description ?

0.5% chance per rank when hit that the damage dealt will be reduced to 1. Does not stack with the equivalent Warpriest effect. This perk has an increased chance to activate on level 20 items.
Release date	25 January 2016Update)
Gizmo type	Armour
Maximum rank	5

Lucky is an Invention fringe benefit which has a 0.5 % prospect per rank when hit that the damage cover will be reduced to 1. It can be created in armor appliance. This perk up does not stack with Saradomin or Zamorak warpriest armor. Lucky works on all damage types, including typeless damage taken in fight. This means that it works on reflected price and Helwyr ‘s bleeds, but does not activate for Araxxor ‘s cling or when trying to eat a dwarven rock cake ( cooled ). [ reservoir needed ]

Reading: Lucky

When the perk up triggers, it prompts the surveil message with X replaced by the decrease : Your Lucky perk up reduces the damage of this attack by X.

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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%)

All numbers in parentheses refer to level 20 gear.

Given a monster ‘s minimal $H_{min}$ and maximum hits $H_{max}$ , and the rank of golden $R$ , the average damage decrease of the perk up over $N$ hits can be calculated approximately as : $0.005NR\times \max \left(\left({\frac {H_{min}+H_{max}}{2}}-1\right),0\right)$

Calculations

The average ratio of damage taken, $r_{avg}$ , from Lucky to that of without Lucky is

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,

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]}}

Notes

A calculator for this is at the top of this page.
Assumption :
- All hits that are taken have the possibility to proc Lucky.
is the rank of the Lucky perk.
$p_{R}$ is the proc chance of Lucky to activate (if the perk is on level 20 gear, then this value is multiplied by 1.1).
$\textstyle \sum _{n=0}^{\infty }\left(1-p\right)^{n}p=1$ when $0\leq p<1$ or $\left|1-p\right|<1$ . This represents summing over all possibilities where integer $n\in [0,\infty )$ represents the amount of hits prior to the proc of Lucky. The probability of Lucky proccing on hit ${n+1}$ is therefore $\left(1-p\right)^{n}p$ .
is the random value uniformly sampled between the enemy’s minimum hit and maximum hit. The describes different sets of hits. The sets are regenerated for every new value of .
- has elements with integer . This is the damage taken before the perk procs.
  - Clarification : This does mean that for $n=0$ that there are no elements in this set.
- $d_{n}^{(1)}$ 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 $d_{n}^{(j)}$ for integer $j\in [0,1]$ is the same in both the numerator and denominator of the ratios for any given .

Simplifications

This can be simplified if the assumption is that every value $\forall$ integer, integer is taken to be the same $\left(d_{n,i}^{(0)}=d_{n}^{(1)}\rightarrow d\right)$ . In this scenario, there is no random element and the above is then only dependent on the proc chance. A table is provided at the top of the page.