# 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 didnt 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%
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%
Sange Lesser Maim Maim Chance: 24%
Heaven's Halberd Lesser Maim Maim Chance: 24%
Sange and Yasha Greater Maim Maim Chance: 30%
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%