Random distribution

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In general Randomness is rejected from competitive gaming since it can decide over loss and defeat without reflecting the actual skill level. Nonetheless there are a couple of random events within Dota 2, that keep the game less predictable and static. To limit the influence of huge streaks, pseudo-random distribution replaces the true random distribution where it made sense for the developers.

Definition[edit]

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

The uniform or true random distribution describes the probability of random event that underlies no manipulation of the chance depending on earlier outcomes. This means that every "roll" operates independently.

The pseudo-random distribution (often shortened to PRD) in Dota 2 refers to a statistical mechanic of how certain probability-based items and abilities work. In this implementation the event's chance increases every time it does not occur, but is lower in the first place as compensation. This results in the effects occurring more consistently.

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 does not, 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.

Example[edit]

Slardar minimap icon.png Slardar's Bash of the Deep icon.png Bash of the Deep has a 25% chance to stun the target. On the first attack, however, it only has 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 of the Deep procs 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. The chance is not reset/increased as long as the ability is on cooldown as well (e.g. Bash).

C Values[edit]

The values of C as a function of nominal chance are given below[1]:

C Nominal Chance Approximate C
0.003801658303553139101756466 5% 0.38%
0.014745844781072675877050816 10% 1.5%
0.032220914373087674975117359 15% 3.2%
0.055704042949781851858398652 20% 5.6%
0.084744091852316990275274806 25% 8.5%
0.118949192725403987583755553 30% 12%
0.157983098125747077557540462 35% 16%
0.201547413607754017070679639 40% 20%
0.249306998440163189714677100 45% 25%
0.302103025348741965169160432 50% 30%
0.360397850933168697104686803 55% 36%
0.422649730810374235490851220 60% 42%
0.481125478337229174401911323 65% 48%
0.571428571428571428571428572 70% 57%
0.666666666666666666666666667 75% 67%
0.750000000000000000000000000 80% 75%
0.823529411764705882352941177 85% 82%
0.888888888888888888888888889 90% 89%
0.947368421052631578947368421 95% 95%

True random events[edit]

Flat distribution[edit]

Some mechanics roll a random value between 0 and 1 and scale it along the minimum / maximum gradient. This includes

  • Chaos Bolt icon.png
    Min Damage: 75/100/125/150
    Max Damage: 200/225/250/275
    Min Stun Duration: 1/1/1/2
    Max Stun Duration: 2/3/4/4
    Damage and stun duration are correlated inversely.
  • Test of Faith icon.png
    Min Damage: 50/100/150/200
    Max Damage: 100/200/300/400
  • Omnislash icon.png
    Min Damage per Slash: 200
    Max Damage per Slash: 225
  • Doppelganger icon.png
    Reappear Radius: 325
    Random positions within the target area. Calculation might be in polar coordinates with 0-360° angle and 0-325 distance units as min/max gradients.
  • Freezing Field icon.png
    Minimum Explosion Distance: 195
    Maximum Explosion Distance: 785
    Angle within the four 90° sectors (0°–90°, 90°–180°, 180°–270°, 270°–360°) may be random.

Discrete distribution[edit]

A couple of random events have discrete states that have similar probability.

  • Powerup Runes with a discrete clamping over the 6 types and 2 locations.
  • Neutral camps can spawn different creeps for each of the small, medium, hard and ancient camps.

Other[edit]

Pseudo random events[edit]

Hero and unit abilities[edit]

Item abilities[edit]

Blind[edit]

Illusory Orb icon.png
Play “We are in accord.”
This page has been marked for an open review.
Please see the discussion pages for additional information, your opinion is welcome.

Since 6.87, Evasion uses pseudo random distribution. It was not mentioned whether Blind and the 25% uphill miss chance are affected by this change. So far no testing has been done.

Legacy data[edit]

The following table shows legacy data cited from the website of the original WC3 DotA. P(T) is the theoretical probability, P(A) is the actual probability. 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. These are higher than those of PRD, hence not as consistent.

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
Version history[edit]
Version
Description

See also[edit]