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Probabilities (6.80)
Danil «Dendi» Ishutin
No luck  only cold heartless planning!This article focuses on the concept and nuances of the pseudorandom distribution.
Pseudorandom distribution in Dota 2
Pseudorandom distribution in Dota 2 (PRD)  is a statistical engine that balances many chancebased abilities to prevent too many or too few activations.
In short, the chance to proc some abilities and items in Dota 2 is not purely random.
Features of PRD
 Mechanism ensures the same result (in this case  effect procs or does not proc) was not repeated too many times in a row.
 The amount of failure can not proc too many times in a row.

In fact, the number of very rare situations is significantly reduced (you need to live a very long life, to see how Phantom Assassin attacks with the Critical Strike 5 times in a row).
Such control of the trigger chance makes the game less dependent on luck and randomness. It reduce unpredictability to almost 0.
The mechanism of the PRD
 When player makes an PRD action there are two scenarios: either the effect triggers, or it does not triggers.
 If effect does not triggered  PRD increases the probability of the effect happening for the next instance by a certain constant.
 If the effect triggered  PRD reduces the probability of the effect happening for the next instance by a certain constant.
 This constant PRD depends on the chance of effect triggering.

The formula for calculating the probability of triggering effect if there are N1 failed attempts:P(N) = C * N
C  constant PRD.
If P(N) is less than 1, effect does not trigger.  When there is a large number of attempts, the chance to trigger the effect is almost equal to the real probability. The constants is chosen in this way.
 So attacking Buildings with critical strike does not increase chance to critical strike on next attack, since critical strike does not work against buildings.

Disadvantage:
For abilities that have a theoretical chance to proc 30% or less, the actual chance to trigger is approximately equal to their theoretical values. Here PRD is almost without discrepancies.
But when abilities have chance to trigger more than 30%, the higher the chance  the greater the difference between the theoretical and actual values of the PRD.
theoretical and actual chance of ability proc  

P(E) %  P(A) %  P(1) %  Theoretical max N  Actual max N  Theoretical C  Actual C 
5  5.0  0.38  263  263  0.00380  0.00380 
10  10.0  1.47  67  67  0.01475  0.01475 
15  15.0  3.22  31  31  0.03222  0.03221 
20  20.0  5.57  17  17  0.05570  0.05570 
25  24.9  8.47  11  11  0.08474  0.08475 
30  29.9  11.89  8  8  0.11895  0.11895 
35  33.6  14.63  6  6  0.15798  0.14628 
40  37.7  18.12  4  5  0.20155  0.18128 
45  41.8  21.87  4  4  0.24931  0.21867 
50  45.7  25.70  3  3  0.30210  0.25701 
55  49.3  29.50  2  3  0.36040  0.29509 
60  53.0  33.32  2  3  0.42265  0.33324 
65  56.6  38.10  2  2  0.48113  0.38109 
70  60.1  42.45  1  2  0.57143  0.42448 
75  63.2  46.13  1  2  0.66667  0.46134 
80  66.7  50.27  1  1  0.75000  0.50276 
 P(E) = probability that is in ability description..
 P(A) = actual probability in the game.
 P(1) = probability of triggerinf effect if there is one attemp in the game.
 Theoretical max N = theoretical number of attempts after which the effect triggers indeed.
 Actual max N = the number of attempts after which the effect triggers indeed.
 Theoretical C = canstant that is used to trigger modifier.
 Actual C = constant, which is used by the game for the necessary calculations.
Types of PRD effects
 Bash.
 Minibash
 Crytical Strike.
 Pierce.
 Damage block.
 Lightning.
 Maim.
List of items and abilities that use PRD
PRD units' abilities  

Hero  Ability 
Theoretical chance to trigger % 
Actual chance to trigger % 
Chance to trigger on the first attack % 
Property of ability 
Brewmaster  Drunken Brawler  10/15/20/25  10/15/20/24.9  1.47/3.22/5.57/8.47  Crytical strike 
Chaos Knight  Chaos Strike 
10 
10  1.47  Crytical strike 
Faceless Void  Time Lock  10/15/20/25  10/15/20/24.9  1.47/3.22/5.57/8.47  Bash 
Juggernaut  Blade Dance 
15/20/25/35 
15/20/24.9/33.6  3.22/5.57/8.47/14.63  Crytical strike 
Lycan  Shapeshift  30  29.9  11.89  Crytical strike 
Phantom Assassin  Coup De Grace 
15 
15  3.22  Crytical strike 
Slardar  Bash 
10/15/20/25 
10/15/20/24.9  1.47  Bash 
Sniper  Headshot  40  37.7  18.12  Bash 
Troll Warlord  Berserker's Rage  10  10  1.47  Bash 
Wraith King  Mortal Strike  15  15  3.22  Crytical strike 
Roshan  Bash  15  15  3.22  Bash 
PRD properties of items  

Item 
Theoretical chance to trigger % 
Actual chance to trigger % 
Chance to trigger on the first attack % 
Property  
Crystalys  20  20  5.57  Crytical strike  
Daedalus  25  24.9  8.47  Crytical strike  
Heaven's Halberd (maim)  15  15  3.22  Maim  
Javelin  20  20  5.57  Pierce  
Maelstrom  25  24.9  8.47  Lightning  
Mjollnir  Static Charge (active) 
20 
20 
5.57  Lightning  
Mjollnir  Chain Lighting (passive)  25  24.9  8.47  Lightning  
Monkey King Bar  35  33.6  14.63  Minibash  
Poor Man's Shield  60  53  33.32  Damage block  
Sange  15  15  3.22  Maim  
Sange & Yasha  16  16  3.69  Maim  
Stout Shield  60  53  33.32  Damage block  
Vanguard  80  66.7  50.27  Damage block 
Examples
Phantom Assassin  Coup De Grace  15% chance to trigger.
The chance to trigger is approximately 3.22% (PRD constant for value 15%).
Every attack without crytical strike increases its chance to 3.22%
 3.22%
 6.44%
 9.66%
 12.88%
 16.1%
 19.32%
 22.54%
etc.
At 30 failures in a row, the 31st time will be surely successfull and the chance for a critical strike again will become equal to 3.22%.
Vanguard  if damage block does not trigger, next attack it will trigger indeed (PRD constant is equal to 50%).
Summary

If modifier (e.g. critical strike) does not trigger, it increases the chance by PRD constant, and if it triggers  the probability becomes equal to PRD constant.
It means that if you hit 5 times in a row with Daedalus and there are no crytical strikes, the probability of the crytical strike, with next 6th hit, is more than 25%(42.5%)
And vice versa, if you have just hit with the crytical strike, the probability to hit with the crytical strike next time is much less than 25%(8.5%)
 Constants are individual for each probability.
 Cases when the effect should not trigger does not change the chance to trigger the next time.

For abilities with theoretical chance of triggering  30% or less, the actual chance is approximately equal to the theoretical value.
And for abilities with P(E) > 30% inaccuracy of triggering will be increasing with the chance growth.
Vanguard's probability of damage block = 66.7% instead of 80%
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