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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. This results in the effects occurring more consistently. In general, PRD is applied to the following types of abilities: Critical Strike, Bash, Damage Block, Chain Lightning, Maim.

Summary[edit]

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

The probability of an effect to occur (or proc) on the Nth test since the last successful proc is given by P(N) = C × N. 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 constant C. This constant, which is also the initial probability, is lower than the listed probability of the effect it is shadowing. Once the effect occurs, the counter is reset. For example, Slardar's Bash has a 25% to Stun the target. On the first attack, however, it will only have an ~8.5% probability to bash. Each subsequent attack without a bash increases the probability by ~8.5%. So on the second attack, the chance is ~17%, on the third it is ~25.5%, etc. After a bash occurs, 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 nearly 25% of the time.

Effects based on PRD rarely proc many times in a row, or go a long time without happening. This makes the game 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.

Aside from damage block items, the probabilities listed in the game are theoretical percentages, which is higher than the actual probability. In the following table, P(T) is the theoretical probability while P(A) is the actual average. C is the PRD constant. Max N is the minimum number of attacks that would result in C × N becoming greater than 1 (i.e. guaranteed proc). Average N is the expected value of N: the sum of the products of N and probabilities. SD is the standard deviation of N, a measure of how spread the data is, using the population formula. The lower the deviation, the more consistent the procs are. For comparison, SDt is the standard deviation of N in true random distribution, using the sample formula from N=1 to N=264. Apparently these are higher than those of PRD, hence not as consistent.

Standard Percentages[edit]

P(T) P(A) C Max N Most Probable N Average N SD SDt
5% 5.0% 0.00380 264 16 20.00 10.30 19.53
10% 10.0% 0.01475 68 8 10.00 5.06 9.50
15% 15.0% 0.03221 32 6 6.67 3.31 6.16
20% 20.0% 0.05570 18 4 5.00 2.43 4.48
25% 24.9% 0.08475 12 3 4.02 1.90 3.49
30% 29.9% 0.11895 9 3 3.34 1.54 2.81
35% 33.6% 0.14628 7 3 2.98 1.35 2.43
40% 37.7% 0.18128 6 2 2.65 1.17 2.10
45% 41.8% 0.21867 5 2 2.39 1.03 1.83
50% 45.7% 0.25701 4 2 2.19 0.91 1.62
55% 49.3% 0.29509 4 2 2.03 0.83 1.45
60% 53.0% 0.33324 4 2 1.89 0.74 1.30
65% 56.6% 0.38109 3 2 1.77 0.69 1.17
70% 60.1% 0.42448 3 2 1.66 0.63 1.05
75% 63.2% 0.46134 3 2 1.58 0.57 0.96
80% 66.7% 0.50276 2 1 1.50 0.50 0.87

Unusual Percentages[edit]

P(T) P(A) C Max N Most Probable N Average N SD SDt
16% 16.0% 0.03645 28 5 6.25 3.09 5.74
17% 17.0% 0.04097 25 5 5.88 2.89 5.37
18% 18.0% 0.04562 22 5 5.56 2.72 5.04
22% 22.0% 0.06668 15 4 4.55 2.19 4.02

List of abilities that use PRD[edit]

Axe - Counter Helix
Counter Helix icon.png
Helix chance: 20%
Actual chance: 20%
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: 20%/25%/30%/35%
Actual chance: 20%/24.9%/29.9%/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
Slow 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]

Crimson Guard - Damage Block
Cost 3850
Crimson Guard icon.png
Chance to block: 80%
Actual chance: 66.7%
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 2800
Maelstrom icon.png
Lightning chance: 25%
Actual chance: 24.9%
Mjollnir - Chain Lightning
Cost 5700
Mjollnir icon.png
Lightning chance: 25%
Actual chance: 24.9%
Mjollnir - Static Charge
Cost 5700
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]