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

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 2016 (Update)
Gizmo type	Armour
Maximum rank	5

Lucky is an Invention perk which has a 0.5% chance per rank when hit that the damage dealt 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 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).

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

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 minimum $H_{min}$ and maximum hits $H_{max}$, and the rank of lucky $R$, the average damage reduction of the perk 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

$r_{avg} = \frac{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum\limits_{i=1}^{n}d^{(0)}_{n,i}\right) + 1 \right]}{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum\limits_{i=1}^{n}d^{(0)}_{n,i}\right) + d^{(1)}_{n} \right]}$

• The average damage reduction is then, after some rearranging,

$1-r_{avg} = \frac{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left(d^{(1)}_{n} - 1 \right)}{\sum\limits_{n=0}^{\infty}\left(1-p_{R}\right)^{n}p_{R}\left[\left(\sum\limits_{i=1}^{n}d^{(0)}_{n,i}\right) + d^{(1)}_{n} \right]}$

Notes

• A calculator for this is at the top of this page.
• Assumption :
  • All hits that are taken have the possibility to proc Lucky.
• $R$ 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 $\left(1-p\right)^{n}p$.
• $d^{(j)}_{n,i}$ is the $i^{th}$ random value uniformly sampled between the enemy's minimum hit and maximum hit. The $j$ describes different sets of hits. The sets are regenerated for every new value of $n$.
  • $d^{(0)}_{n,i}$ has $n$ elements with integer $i \in [1,n]$. 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.
  • $d^{(1)}_{n}$ 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 $d^{(j)}_{n}$ for integer $j \in [0,1]$ is the same in both the numerator and denominator of the ratios for any given $n$.

Simplifications

• This can be simplified if the assumption is that every value $d^{(0)}_{n,i}$ $\forall$ integer $i \in [1,n]$, $d^{(1)}_{n}$ $\forall$ integer $n \in [0,\infty)$ 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 $r_{avg}$ 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

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

Sources

Material		Rarity	Perk ranks with X materials
			1	2	3	4	5
	Silent components	Rare	1	2	3	4–5	5
	Delicate parts	Common	0	0	1	1–2	1–2
	Light components	Uncommon	0	1–2	1–2	1–3	1–5
	Fortunate components	Rare	1	1–2	1–3	1–5	1–5

Suggested gizmos

Gizmo layout	Possible perks
	Lucky 5
	Lucky 5 Hoarding Other possible perks: Brassican
	Lucky 4–5 (5 is rare) Mobile Other possible perks: Crystal Shield Demon Bait
	Lucky 4–5 (5 is rare) Mobile Other possible perks: Reflexes Dragon Bait
	Crackling 2–3 (3 is rare) Lucky 1 Other possible perks: Ultimatums