Idea : Danil «Dendi» Ishutin
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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 pseudo-random distribution.
Danil «Dendi» Ishutin

Pseudo-random distribution in Dota 2

Pseudo-random distribution in Dota 2 (PRD) - is a statistical engine that balances many chance-based 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 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

  1. When player makes an PRD action there are two scenarios: either the effect triggers, or it does not triggers.
  2. If effect does not triggered - PRD increases the probability of the effect happening for the next instance by a certain constant.
  3. If the effect triggered - PRD reduces the probability of the effect happening for the next instance by a certain constant.
  4. This constant PRD depends on the chance of effect triggering.
  5. The formula for calculating the probability of triggering effect if there are N-1 failed attempts:
    P(N) = C * N
    C - constant PRD.
    If P(N) is less than 1, effect does not trigger.
  6. 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.
  7. So attacking Buildings with critical strike does not increase chance to critical strike on next attack, since critical strike does not work against buildings.
  8. 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.
  • Mini-bash
  • 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 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 Knight  Chaos Strike

10

10 1.47 Crytical strike
faceless_void Faceless Void  Time Lock 10/15/20/25 10/15/20/24.9 1.47/3.22/5.57/8.47 Bash
juggernaut Juggernaut  Blade Dance

15/20/25/35

15/20/24.9/33.6 3.22/5.57/8.47/14.63 Crytical strike
lycan Lycan  Shapeshift 30 29.9 11.89 Crytical strike
phantom_assassin Phantom Assassin  Coup De Grace

15

15 3.22 Crytical strike
slardar Slardar  Bash

10/15/20/25

10/15/20/24.9 1.47 Bash
sniper 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 Mini-bash
 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 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%

  1. 3.22%
  2. 6.44%
  3. 9.66%
  4. 12.88%
  5. 16.1%
  6. 19.32%
  7. 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%
AuthorJane Lovejoy Date 3 March 2014, 16:43 Views14894 Comments13
Comments (13)
trixilon
Gold 18 Respect 14
DOTA
#1 tr trixilon 20 June 2014, 10:51
I wasnt know that... Thanks for that mate ! )
0
Oblivion
Gold 35 Respect 1
#2 ua Oblivion 30 June 2014, 11:01
As usual, Na'vi's guide is throughout and comprehensive...
+1
User Deleted
#3 1 July 2014, 08:33
I just realized it wasn't luck at all!
0
Stenskott
Gold 0 Respect 0
DOTA
#4 se Stenskott 31 July 2014, 19:07
Really nice guide. Very well written and easy to understand!
0
hryniel
Gold 20 Respect 4
#5 hu hryniel 9 August 2014, 20:11
Nice guide! Thanks!
0
edrilordz
Gold 455 Respect 1276
#6 ph edrilordz 2 September 2014, 05:28
NICE NICE NICE ^_^
0
builder_247
Gold 9 Respect 1
#7 fi builder_247 23 September 2014, 18:00
Awesome Dendi guide as usual, good luck for tournaments!
0
goiastrolol
Gold 0 Respect 0
#8 br goiastrolol 25 September 2014, 21:23
Awesome Guide Thank you!
0
edrilordz
Gold 455 Respect 1276
#9 ph edrilordz 1 October 2014, 07:52
very well said!!!
0
edrilordz
Gold 455 Respect 1276
#10 ph edrilordz 1 October 2014, 07:52
thank NaVi!!!
0
J4ckTh3R1pp3r
Gold 0 Respect 1
DOTA
#11 it J4ckTh3R1pp3r 19 December 2014, 00:51
Interesting Guide for PRD!!! Well done guys for who don't know this parameter or calc :)
0
#12 ph ciel040612 26 March 2015, 04:17
This is good information. Now I want to play sniper, juggernaut and faceless void more! ^^ thank you NAVI!
0
Aenish
Gold 0 Respect 0
DOTA
#13 in Aenish 26 April 2016, 12:17
As u know the one and only navi guide is awesome and even in game i use navi guide
Thanks for it guys I love you all
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