Random distribution
Randomness is sometimes rejected from competitive gaming since it can decide over loss and defeat without reflecting the actual skill level. However, coping with the unexpected is an aspect of skill. 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, pseudorandom distribution replaces the true random distribution where it made sense for the developers.
Contents
Definition[edit]
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 pseudorandom distribution (often shortened to PRD) in Dota 2 refers to a statistical mechanic of how certain probabilitybased 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.
Technically, if X_i corresponds to the event occurring on trial i and ~X_i is the inverse of that event, pseudorandom distribution enforces that P(X_N  ~X_i for all i < N) = C × N. Also, when C × N ≥1, which occurs on the first trial N where N ≥ 1/C, the event is guaranteed if it has not already happened. This results in a distribution where P(X_k) = (1C)(12C)...(1(k1)C)KC = k!C Π_{i=1}^{k1} (1/i  C).
Example[edit]
 On melee heroes, Skull Basher's Bash 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 effect 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 probabilitybased 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
 Attack damage of units, if they have a minimum and maximum attack damage range.
 Last hit gold bounty, if a minimum and maximum range is given.
 Roshan's respawn timer.
 Reappear Radius: 325
Random positions within the target area. Calculation might be in polar coordinates with 0360° angle and 0325 distance units as min/max gradients.  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.
 The position of the caster and the illusions within the formation is random.
 Attack Radius: 300
Attack Interval: 0.25 sec  Number of Bounces: 3/4/5/6
Bounce Distance: 475
Although prioritizing heroes over units, and injured allies over noninjured ones, if multiple units meet the criteria, it chooses between them randomly.  Search Radius: 700
Spirits choose their targets randomly. Once chosen, they stick with that target until they can't attack it anymore. The spirits prioritise units the caster attacks.  Max Targets: 2
Search Radius: 400  Effect Radius: 250/350/450/550
Attack Interval: 0.2 sec
Targets are determined upon cast. The order of attacks is random.  Rockets per Second: 10
Search Radius: 400
Interval: 0.1
Duration: 3  Number of Bounces: 4
Bounce Distance: 500
Interval: 0.25 sec  Attack Rate Divisor: 1.7
Search Radius: 425
Slashes enemies based on Juggernaut's attack rate.  Search Radius: 750
Interval: 1.75
Requires Aghanim's Scepter. When upgraded, strikes a random enemy within the radius with a single Lightning Storm bolt.  The position of the caster and the illusions within the formation is random.
 Max Targets: 1 ( 4)
Search Radius: 900/1000/1100/1200
Requires a talent. When upgraded, releases three extra projectiles, hitting a second, third and fourth random enemy within the search radius.  Max Targets: 1 ( 4)
Search Radius: 900/1000/1100/1200
Requires a talent. When upgraded, releases three extra projectiles, hitting a second, third and fourth random enemy within the search radius.  The position of the caster and the illusions within the formation is random.
 Multicast Amount: 1/2/3
Multicast Range: 1400
Interval: 0.4 sec  Multicast Amount: 1/2/3
Search Radius: 700  Max Targets: 1 ( 3)
Search Radius: 825/1050/1275/1500
Requires a talent. When upgraded, releases two extra projectiles, hitting a second and third random enemy within the search radius.  Search Radius: 500
When multiple enemies have the same amount of health, or if multiple heroes are linked with Static Link, chooses between them randomly.  Max Targets: 1 ( 2)
Search Radius: 0 ( 875)
Requires Aghanim's Scepter. When upgraded, releases a second attack on each attack, hitting a second random enemy within the search radius.  Max Targets: 1 ( 2) ( 2, 3)
Search Radius: 0 ( 700) ( 700)
Requires a talent or Aghanim's Scepter. When upgraded, releases a second/third projectile, hitting a second/third random enemy within the search radius.  Max Targets: 1 ( 2)
Search Radius: 0 ( 700, Global)
Requires Aghanim's Scepter. When upgraded, releases a second projectile, hitting a second random enemy within the search radius.  Max Targets: 1 ( 2)
Search Radius: 0 ( 700)
Requires Aghanim's Scepter. When upgraded, targets a second random enemy within the search radius.  Number of Flares: 1 ( 2)
Search Radius: 0 ( 700)
Requires Aghanim's Scepter. When upgraded, creates a second flare on a random enemy within the search radius.  Max Targets: 1 ( 2)
Search Radius: 1200 ( 1325)
Requires a talent. When upgraded, releases a second projectile, hitting a second random enemy within the search radius.
Other[edit]
Pseudo random events[edit]
Along with the listed abilities, the 25% uphill miss chance and the drop chance of neutral items use pseudo random distribution as well.
Hero and unit abilities[edit]
 Critical Chance: 20%
 Helix Chance: 17%/18%/19%/20%
 Critical Chance: 80%
Evasion: 50%/60%/70%/80%  Miss Chance: 30%/40%/50%/60%
 Proc Chance: 12%/16%/20%/24%
 Proc Chance: 20%
 Critical Chance: 20%/25%/30%/35%
 Miss Chance: 70%
 Cripple Chance: 20%
 Proc Chance: 17%
 Proc Chance (Hero): 40%/45%/50%
Proc Chance (Illusion): 8%  Critical Chance: 30%
 Miss Chance: 25%/35%/45%/55%
 Bash Chance: 15%
 Slow Chance: 40%
 Proc Chance: 20%
 Root Chance: 14%/16%/18%/20%
 Miss Chance: 60%
 Critical Chance: 9%/11%/13%/15%
Item abilities[edit]
 Bash Chance (Melee/Ranged): 25%/10%
 Block Chance: 50%
 Evasion: 35%
 Block Chance: 50%
 Critical Chance: 30%
 Critical Chance: 30%
 Evasion: 25%
 Pierce Chance: 25%
 Lightning Chance: 25%
 Lightning Chance: 25%
 Lightning Chance: 20%
 Pierce Chance: 75%
 Miss Chance: 17%
 Bash Chance (Melee/Ranged): 25%/10%
 Evasion: 15%
 Block Chance: 50%
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]
 Blind now uses pseudorandom distribution. (Uphill miss chance, Incapacitating Bite, Blinding Light, Burn, Sand Storm, Smoke Screen, Whirling Axes (Melee))
 Cinder Brew's selfattack chance now uses pseudorandom distribution.
 Shapeshift's critical strike now uses pseudorandom distribution.
 Walrus PUNCH! passive trigger talent now uses pseudorandom distribution.
 Juxtapose now uses pseudorandom distribution.
 Craggy Exterior now uses pseudorandom distribution.
 Alpha Wolf's Critical Strike now uses pseudorandom distribution.
 Giant Wolf's Critical Strike now uses pseudorandom distribution.
 Essence Aura now uses pseudorandom distribution.
 Evasion now uses pseudorandom distribution.
 Entangling Claws now uses pseudorandom distribution.
 Greater Bash now uses pseudorandom distribution.
 Counter Helix now uses pseudorandom distribution.
 Moment of Courage now uses pseudorandom distribution.
References[edit]
