Pseudo-random distribution

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Not to be confused with pseudorandom number generation (PRNG), an algorithm for generating a sequence of numbers that approximates the properties of random numbers.

The Pseudo-random distribution (often shortened to PRD) in DotA refers to the statistical mechanics of how certain probability-based items and abilities work. In true random distribution, every "roll" operates independently, but in PRD, the effect's chance increases every time it does not happen. In general, PRD is applied to the following types of abilities: Critical Strike, Bash, Damage Block, Chain Lightning, Maim.


The effects of PRD on the number of attacks until the next proc.

For each instance which could trigger the effect but doesn't, the PRD augments the probability of the effect happening for the next instance by a certain constant. This constant (which is also the initial probability) is usually quite low compared to the stated probability of the effect it is shadowing. This probability counter resets every time an instance of the effect occurs. Over a moderately large period of time, the expected probability for each instance is almost exactly the listed probability (but see the note below), and since the probability increases steadily for each time the effect doesn't happen (and resets when it does happen), the effect occurs more consistently.

The important gameplay and balance effect of PRD is that effects based on it rarely occur many times in a row, or go a long time without happening. In the case of Bash, a bash will occur very nearly every four attacks, will rarely occur twice in a row, and will always happen within 12 attacks. This makes the game far less luck based and adds a great deal of consistency to many probability based abilities in Dota 2.

Gameplay wise, PRD is difficult to exploit. It is theoretically possible to increase your chance to bash or critical strike on the next attack by attacking creeps several times without the effect happening, but in practice this is nearly impossible to do. Note that for instances that would not trigger the effect, the probability counter does not increase. So a hero with critical strike attacking Buildings does not increase its chance to critical strike on its next attack, since critical strike does not work against buildings.

The probability of a modifier occurring on the Nth attack since the last successful modifier is given as P(N) = C * N. C is a constant derived from the expected probability of the modifier occurring. C serves as both the initial probability of the modifier and the increment by which it increases every time the modifier fails. When P(N) becomes greater than 1.00, the modifier will always succeed.

For example, Slardar's Bash has a 25% probability to Stun the target. On the first attack, however, it will only have an ~8.5% probability to bash; this is its PRD constant C. Each subsequent attack without a bash increases this probability by 8.5%. So on the second attack, the chance is 17%, on the third it is 25.5%, etc. After a bash procs, the probability resets to 8.5% for the next attack. These probabilities average out so that, over a moderate period of time, Bash will proc very nearly 25% of the time.

However, the table of C values used by Dota 2 will not always result in a modifier's listed chance being equal to its actual probability of occurring. This is especially clear with chances over 25%. For example, a Vanguard has a listed chance of 80% to block damage. To achieve that probability using PRD, a C value of around 75% would be used. However, the constant used by the game is closer to 50%, resulting in the actual chance to block being around 66%.

In the following table, P(T) is the theoretical probability of the modifier occurring. In the case of Bash, this is 25%. P(A) is the actual probability of the modifier occurring over an infinite number of attacks. Theoretical C is the constant that would result in the theoretical probability, while Actual C is the constant used by the game, resulting in the skewed probability. Max N is the minimum number of attacks that would result in C * N becoming greater than 1.00, and the maximum number of attacks that could possibly occur before a modifier succeeds.

Standard Percentages[edit]

P(T) P(A) Theoretical C Actual C Theoretical Max N Actual Max N
5% 5.0% 0.00380 0.00380 264 264
10% 10.0% 0.01475 0.01475 68 68
15% 15.0% 0.03222 0.03221 32 32
20% 20.0% 0.05570 0.05570 18 18
25% 24.9% 0.08474 0.08475 12 12
30% 29.9% 0.11895 0.11895 9 9
35% 33.6% 0.15798 0.14628 7 7
40% 37.7% 0.20155 0.18128 5 6
45% 41.8% 0.24931 0.21867 5 5
50% 45.7% 0.30210 0.25701 4 4
55% 49.3% 0.36040 0.29509 3 4
60% 53.0% 0.42265 0.33324 3 4
65% 56.6% 0.48113 0.38109 3 3
70% 60.1% 0.57143 0.42448 2 3
75% 63.2% 0.66667 0.46134 2 3
80% 66.7% 0.75000 0.50276 2 2

Unusual Percentages[edit]

P(T) P(A) Theoretical C Actual C Theoretical Max N Actual Max N
16% 16.0% 0.03645 0.03645 28 28
17% 17.0% 0.04097 0.04097 25 25
18% 18.0% 0.04562 0.04562 22 22
22% 22.0% 0.06668 0.06668 15 15

List of abilities that use PRD[edit]

Axe - Counter Helix
Counter Helix icon.png
Helix chance: 17%
Actual chance: 17%
Brewmaster - Drunken Brawler
Drunken Brawler icon.png
Critical chance: 10%/15%/20%/25%
Actual chance: 10%/15%/20%/24.9%
Chaos Knight - Chaos Strike
Chaos Strike icon.png
Critical chance: 10%
Actual chance: 10%
Faceless Void - Time Lock
Time Lock icon.png
Bash chance: 10%/15%/20%/25%
Actual chance: 10%/15%/20%/24.9%
Juggernaut - Blade Dance
Blade Dance icon.png
Critical chance: 15%/20%/25%/35%
Actual chance: 15%/20%/24.9%/33.6%
Legion Commander - Moment of Courage
Moment of Courage icon.png
Lifesteal Chance: 16%/18%/20%/22%
Actual Chance: 16%/18%/20%/22%
Lycan - Shapeshift
Shapeshift icon.png
Critical chance: 30%
Actual chance: 29.9%
Phantom Assassin - Coup de Grâce
Coup de Grace icon.png
Critical chance: 15%
Actual chance: 15%
Roshan - Bash
Roshan bash icon.png
Bash chance: 15%
Actual chance: 15%
Slardar - Bash
Bash icon.png
Bash chance: 10%/15%/20%/25%
Actual chance: 10%/15%/20%/24.9%
Sniper - Headshot
Headshot icon.png
Bash chance: 40%
Actual chance: 37.7%
Troll Warlord - Berserker's Rage Active
Berserker's Rage Active icon.png
Bash chance: 10%
Actual chance: 10%
Wraith King - Mortal Strike
Mortal Strike (Wraith) icon.png
Critical chance: 15%
Actual chance: 15%

List of items that use PRD[edit]

Crystalys - Critical Strike
Cost 2150
Crystalys icon.png
Critical chance: 20%
Actual chance: 20%
Daedalus - Critical Strike
Cost 5550
Daedalus icon.png
Critical chance: 25%
Actual chance: 24.9%
Heaven's Halberd - Lesser Maim
Cost 3850
Heaven's Halberd icon.png
Maim chance: 15%
Actual chance: 15%
Javelin - Pierce
Cost 1500
Javelin icon.png
Pierce chance: 20%
Actual chance: 20%
Maelstrom - Chain Lightning
Cost 2700
Maelstrom icon.png
Lightning chance: 25%
Actual chance: 24.9%
Mjollnir - Chain Lightning
Cost 5600
Mjollnir icon.png
Lightning chance: 25%
Actual chance: 24.9%
Mjollnir - Static Charge
Cost 5600
Mjollnir icon.png
Lightning chance: 20%
Actual chance: 20%
Monkey King Bar - Mini-bash
Cost 5400
Monkey King Bar icon.png
Chance to mini-bash: 35%
Actual chance: 33.6%
Poor Man's Shield - Damage Block
Cost 550
Poor Man's Shield icon.png
Chance to block non-hero damage: 60%
Actual chance: 53%
Sange - Lesser Maim
Cost 2050
Sange icon.png
Maim chance: 15%
Actual chance: 15%
Sange and Yasha - Greater Maim
Cost 4100
Sange and Yasha icon.png
Maim chance: 16%
Actual chance: 16%
Stout Shield - Damage Block
Cost 250
Stout Shield icon.png
Chance to block: 60%
Actual chance: 53%
Vanguard - Damage Block
Cost 2225
Vanguard icon.png
Chance to block: 80%
Actual chance: 66.7%

See Also[edit]