Pseudo Random Distribution

As chance would have it.

Pseudo Random Distribution (shortened PRD) is a mechanic ported to Dota 2 which originated from the Warcraft 3 engine. Pseudo-random distributions are used to calculate chance based skills including critical strike, bash, and evasion abilities. In more common random distributions, every possible check has the same chance of activation regardless of any prior checks. In pseudo-random distribution streaks of activation or streaks of no activation are less likely, but the overall probability remains the same.

Summary

The PRD works by increasing the percentage chance for an activation by a certain amount every time it didn't activate until the chance for activation is over 100%. On the other side right after an activation the chance for another activation is significantly lower than the chance stated in the skill/item description. This means that skills such as critical strike will occur more consistently than they would in more common standard random distributions. Significantly streaks of "good luck" or "bad luck" are less likely to occur, making the game less luck dependent.

The base and incremental chances were precalculated and accessed from table because the calculations needed to produce those chances for each probability under PRD are taking up much more processor runtime than those for the same probability under standard random distribution. The values in Warcraft 3 were in 5% steps and, curiously, incorrect for chances of 15% and higher. These errors increase for each step and become noteworthy from 35% onward. It is assumed that these errors were ported over to Dota 2 for parity reasons.

The chance for an activation on first hit/check is also the amount by which the chance is incremented each time no activation occurs. The chance will not increase by an event in which an activation cannot occur. This makes abusing the PRD difficult, because, for example, hitting a building will not increase the chance of a critical strike, because you critical strike do not work against buildings in the first place.

In the following table, P(E) is the probability that the PRD is trying to mimic. Theoretical C is the value the PRD table should use to result in the correct probability. Actual C is the value used by the Warcraft 3 engine (and presumably Dota 2), resulting in a skewed probability. Max N is the number of attacks after which the chance for an activation(C*N) reaches 100%. P(A) is the actual probability of an activation occurring.

P(E)	Theoretical C	Actual C	Theoretical Max N	Actual Max N	P(A)
5%	0.00380	0.00380	263	263	5.0%
10%	0.01475	0.01475	67	67	10.0%
15%	0.03222	0.03221	31	31	15.0%
20%	0.05570	0.05570	17	17	20.0%
25%	0.08474	0.08475	11	11	24.9%
30%	0.11895	0.11895	8	8	29.9%
35%	0.15798	0.14628	6	6	33.6%
40%	0.20155	0.18128	4	5	37.7%
45%	0.24931	0.21867	4	4	41.8%
50%	0.30210	0.25701	3	3	45.7%
55%	0.36040	0.29509	2	3	49.3%
60%	0.42265	0.33324	2	3	53.0%
65%	0.48113	0.38109	2	2	56.6%
70%	0.57143	0.42448	1	2	60.1%

The following graphics show how the distribution for a 15% chance differ between PRD and standard random distribution.

Abilities that use PRD

Hero Abilities
Hero	Ability	Chance
Axe	Counter Helix	Counter Helix Chance: 20%
Brewmaster	Drunken Brawler	Critical Strike Chance: 10% / 15% / 20% / 25% Evasion Chance: 10% / 15% / 20% / 25%
Chaos Knight	Chaos Strike	Critical Strike Chance: 10%
Faceless Void	Time Lock	Bash Chance: 10% / 15% / 20% / 25%
Juggernaut	Blade Dance	Critical Strike Chance: 20% / 25% / 30% / 35%
Legion Commander	Moment of Courage	Moment of Courage Chance: 25%
Lycan	Shapeshift	Critical Strike Chance: 40%
Outworld Devourer	Essence Aura	Mana Restore Chance: 40%
Phantom Assassin	Blur	Evasion Chance: 20% / 30% / 40% / 50%
	Coup de Grace	Critical Strike Chance: 15%
Phantom Lancer	Juxtapose	Hero Chance: 40% / 45% / 50% Illusion Chance: 8%
Slardar	Bash of the Deep	Bash Chance: 10% / 15% / 20% / 25%
Sniper	Headshot	Headshot Chance: 40%
Spirit Breaker	Greater Bash	Bash Chance: 17%
Tiny	Craggy Exterior	Stun Chance: 25%
Troll Warlord	Berserker's Rage	Bash Chance: 10%
Wraith King	Mortal Strike	Critical Strike Chance: 15%
Unit Abilities
Unit	Ability	Notes
Lycan Wolf	Cripple	Cripple Chance: 20%
Giant Wolf	Critical Strike	Critical Chance: 20%
Alpha Wolf	Critical Strike	Critical Chance: 20%
Roshan	Bash	Bash Chance: 15%
Item Abilities
Item	Ability	Notes
Stout Shield	Damage Block	Damage Block Chance: 50%
Vanguard	Damage Block	Damage Block Chance: 50%
Crimson Guard	Damage Block	Damage Block Chance: 50%
Abyssal Blade	Damage Block	Damage Block Chance: 50%
Poor Man's Shield	Damage Block	Creep Damage Block Chance: 50%
Monkey King Bar	Mini-Bash	Mini-Bash Chance: 35%
Crystalys	Critical Strike	Critical Strike Chance: 20%
Daedalus	Critical Strike	Critical Strike Chance: 30%
Skull Basher	Bash	Melee Hero Bash Chance: 25% Ranged Hero Bash Chance: 10%
Abyssal Blade	Bash	Melee Hero Bash Chance: 25% Ranged Hero Bash Chance: 10%
Maelstrom	Chain Lightning	Chain Lightning Chance: 25%
Mjollnir	Static Charge	Static Charge Lightning Chance: 20%
	Chain Lightning	Chain Lightning Chance: 25%
Talisman of Evasion	Evasion	Evasion Chance: 20%
Heaven's Halberd	Evasion	Evasion Chance: 25%
Solar Crest	Evasion	Evasion Chance: 25%
	Shine	Evasion Chance: 25%
Butterfly	Evasion	Evasion Chance: 35%

External Links

• [Article] Pseudo-Random Distribution - DotA Forums