Pseudo-Random Mechanics - And How to Use Them To Your Advantage
Pseudo is a funny word. It means having the appearance of something without having the content. You’ve heard of a pseudo-intellectual, or somebody who pretends to know more than he does for status.
In Dota, unreliable isn’t always random.
Pseudo-random mechanics are those which appear unpredictable, but in reality have manipulated odds which can be abused. Critical strikes, bashes, and maims all use what is called pseudo-random distribution (PRD) instead of true-random distribution.
The impact? These abilities are more likely to activate if they have not done so in a while, and less likely to activate if they did so recently. PRD is like trying to pull the Ace of Spades from a deck of 52 cards. Everytime you draw a wrong card you remove it, reducing the amount of cards left in the deck. If you've missed fifty times, there are only two cards left. Your odds on future success go up every time you miss.
Let’s take 
Juggernaut's 
Blade Dance as an example. Every time Juggernaut makes an attack against a valid target, it increases the likelihood that his ability will activate (or proc) on the following attempt.
Even though the odds of achieving a critical strike are listed as 35% at level four, the chances of getting a proc on the first try are lower. Each missed attempt will increase the odds until the ability activates, at which point it resets. So a level four Blade Dance will be have a 14% chance to activate on the first attack since the counter reset, but a 43% chance to activate on the fourth attack. It doesn’t matter how many different units Juggernaut targets or how long he waits between attacks. Juggernaut can rely more heavily on a critical if he hasn’t seen one in a while and is guaranteed a proc every seven attempts.
True random doesn't work that way. Remember 
Spirit Breaker's 17% bash? Until recently, that was the only bash in the game that wasn’t Pseudo-Random. This meant you could never rely on a bash, but also that chains of bashes were much more likely.
The odds of getting a bash on three consecutive, applicable attacks were about 1:200. Now, Spirit Breaker uses Pseudo-random on 
Greater Bash. With PRD, the odds would be closer to 1:10000. It makes a big difference. The goal of this design is to prevent players from being advantaged or disadvantaged by pure luck. PRD tempers the impacts of chance in the game.
What Uses PRD?
Basically all non-guaranteed offensive abilities use Pseudo-Random: critical strikes, bashes, maims, damage return such as 
Axe’s 
Counter Helix or 
Legion Commander’s 
Moment of Courage, or attack modifiers such as 
Maelstrom’s Chain Lightning. In addition, damage block uses Pseudo-Random. That isn’t particularly useful to know, though.
True random is used for miss chance (evasion, blind, uphill attacks) and for random events including Omnislash targets, Ogre Magi’s Multicast, or rune spawning. Non-hero abilities also use true random, meaning the critical strike on summoned or neutral units, or 
Lone Druid’s Spirit Bear’s Entangling Claws.
Generally, if it adds effects to an attack, it uses PRD. If it creates a miss or modifies a casted spell, it uses true random.
How Can It Give Me an Advantage?
There are a shocking number of ways understanding this mechanic can give you an immense advantage. Let’s start with choosing your laning build.
Laning
The second column is the probability of getting a
Bash if you are on a specific attack. The third column
show the cumulative chances, or the probability of
getting a bash on that attack or any attack prior
Consider 
Slardar. A mere 10% chance of a [missing skill: slardar-bash-5116] does not seem worthwhile in lane, and it wouldn’t be at pure random. As a friend of mine said recently, 10% odds of success means 90% odds of irrelevance. But with PRD, that’s not true. Slardar can count his number of non-bashes. He can count those while he is attacking creeps in order to increase his chances of a bash on a hero later on. This means he has a much greater than 10% chance to bash if he’s savvy.
At level four, he gets three free attacks through the duration of 
Slithereen Crush.
With Slardar’s Bash, the odds of at least one bash in the first three attacks are only about one in thirteen. However, the odds of at least one bash in his fifth, sixth, and seventh attacks are one in four. By counting non-bashes until it amounts to seven attacks, Slardar can increase his chances of bashing during Slithereen Crush from 8% to 35%. That’s with only a level one bash.
This principle is true for every hero with an innate attack modifier that uses PRD. A casual point in any bash, maim, or critical strike can be leveraged to increase gank success. Supports should be aware of this principle and help time gank attempts for when bashes or maims are more likely. If you are in a 1v1 lane, a point in a PRD skill is usually worthwhile if you stack the deck in your favor, choosing to harass when your attacks are more likely to be effective.
This is especially true for 
Sniper. Where possible, Sniper should harass if 
Headshot has missed two attacks in a row: the majority of the time, he will proc a headshot on the third attack, and he is guaranteed a headshot by the sixth attack. Since this ability can activate on allies, this makes spamming attacks on allied creeps particularly effective. This way you are constantly counting up damage potential without pushing out your lane.
Animation Priming
Sometimes cancelled attacks will queue up an ability proc, but the attack is cancelled before it hits. Whenever that animation begins, it resets the PRD counter. This is especially visible on heroes with critical hits such as 
Phantom Assassin’s 
Coup de Grace or 
Chaos Knight’s 
Chaos Strike.
The obvious initial reaction is that this isn’t fair: you didn’t attack, but now future attacks are less likely to have your ability. That’s like a bus driver taking your fare but making you walk.
“ ...every cancelled attack that is not a proc adds to the counter”
The plus side is that every cancelled attack that isn’t a proc adds to the counter. Most PRD critical strikes have sounds and custom animations associated with them, making it easy to tell if you cancelled one. You can use animation cancelling to prime your critical strike. Let’s take Phantom Assassin as an example. If you are among the majority of players who opt to build Battle Fury on this hero, you can increase your jungle farming rate in the early and mid game. Every time you critical strike, simply cancel four attacks as fast as you can. It should take, in total, less than a second, but you will increase the rate at which you critical strike. Yes, you will cancel some critical strikes with this technique, but you will see more than twice as many as you cancel. Coup de Grace’s probability for a proc in the first four attacks is 28.7%, compared to 61% in the following four attacks.
Teamfight and Gank Windows
The same idea can be used prepare for teamfights. If you count any PRD ability out before starting a teamfight, you’ll increase your odds of getting multiple procs while fighting. For example, if you manage to have six regular attacks as Phantom Assassin prior to a teamfight, your odds of getting a Coup de Grace proc during 
Phantom Strike increase to more than seven in ten. You are also about five times as likely to get two crits during those attacks. Remember the Spirit Breaker change I mentioned earlier? Consider this: if you charge after failing to bash on six attacks, you are almost guaranteed to chain a bash during your gank. Neither 
Charge of Darkness nor 
Nether Strike resets to counter for Greater Bash.
You should actively prepare the Greater Bash counter before charging to have a stun from charge, a stun from Nether Strike, and two to free attacks after Nether Strike which are about seventy percent likely to proc a Greater Bash for a third stun. Especially for a support Spirit Breaker, this is crucial to increasing gank success in the early game.
Best of all, your opponents will never be able to know when you’ve primed your hero for an engagement because you can do so in the fog of war.
Trolling with Towers
PRD abilities only count up for attacks which could proc their effects. This means that you cannot gain or lose odds of a bash or critical strike while attacking a structure. One way to force favorable fights is to siege a tower while holding onto high PRD odds. If you switch to a hero, you are already prepared to proc your effects. Attacking creeps will invalidate this strategy. It works best with heroes who tank towers effectively, such as Wraith King or Juggernaut.
Another way to abuse this mechanic is to use 
Troll Warlord’s 
Berserker's Rage to store bash odds. At 10%, the average probability for stunning with Troll Warlord is fairly low. However, his ranged attacks do not impact the counter. If you attack creeps at melee range without bashing, switch to range. You can continue to attack normally on any unit without resetting your counter. Then switch to melee when you think a bash could lead to a kill.
If you store after ten missed attacks, a third of attempts you will bash within two attacks of when you switch back. That means you can continue to lane and will still be able to store a likely bash.
Dirty.
Limitations
Even manipulating PRD, you’ll still miss proc opportunities. Just because something is more likely does not mean it will always happen. Even so, a difference between engaging with a one in twenty chance to proc and a one in three chance is not easily ignored. You should not hold onto PRD counters at the expense of creep kills unless you expect an imminent opportunity to engage. You can always store up PRD advantage again.
Imagine you have a support hiding and your opponent is stepping out of place. In this case, it’s worth missing one or two creep kills to significantly increase your odds of killing the hero by preserving your favorable PRD counter. You also don’t usually want to pick up value points in PRD abilities if you are in a difficult lane (for example, Slardar offlane against a trilane). The more difficult your lane, the less likely you will be to find a good kill opportunity. If kill opportunities are not lining up with when your PRD is stacked, then you’ve wasted the skill point entirely.
Rules of Thumb
The more misses you’ve racked up, the better it is to attack a hero. Random attack modifiers typically use PRD. Obviously reaching a high number of misses becomes increasingly less likely with every attack, because if you do proc your effect the counter resets. There’s a balance between trying to reach a realistic number of missed chances and still being able to expect a hit soon.
10% skills are most realistically best-saved until nine misses, which should happen about half of chains. If you try to wait for more than nine misses, you will fail on a majority of attempts and if you don’t wait until nine misses you are not maximizing your returns.
15% skills are best saved until six misses. 25% skills are best saved until three misses. 40% skills such as Headshot typically aren’t worth saving at all because they are so frequent, but if you do reach two misses, it is worthwhile to harass the nearest opponent to increase hero damage output. Just don’t put yourself in danger to do so.
Gorgon the Wonder Cow is a freelance Dota writer and esports analyst. You can find him at twitter.com/theWonderCow.
Image Source: Liquipedia
Update:
The table for Slardar and the cumulative odds of a Bash was initially wrong and has since been corrected.
@Cunnilinguistician: It's deliberately designed that way. In normal RNG which mimics real random, each random event occurs INDEPENDENTLY. Which means, if that's the case, there is chance that Phantom assassin will not have a single crit in the entire game (very unlikely, probability is very close to 0, but possible). There is also chance that he will crit every single attack (very unlikely too, but also possible). This situation leaves the result to luck: In a combat, PA can crit many times and rekt enemy team, or no crit at all and meet her doom. They don't want it happens that way. They want luck to be manageable.
So, the solution is, there is a counter which counts PA attacks and reset when she crit. The first attack of counter will have very low crit chance, with next attack have higher crit chance. The crit chance reach 100% at 17 consecutive no-crit attack (if i'm not mistaken). PA will get 15% crit in the long run, when number of attack reach infinity. Therefore, crit chance is manageable as you see in the article.
You are perfectly true in thinking there is correlation, cause crit chance of each attack is not independent. Therefore it's deliberatively called pseudo-random, not real random or just rng or something.
This mechanics stayed with the game since beginning of dota 1, and they recreated it in dota 2 also. The method of abuse in this article is not new at all, just old knowledge. You can search for many documents describe this in detail, not only this patch and not only dota 2. Here is one of them: http://dota2.gamepedia.com/Pseudo-random_distribution. Some people say we can look at the source code of dota 1 map (Warcraft 3 custom map) and find detail function of the pseudo rng. I have never do it myself, because I dont understand code.
You can do experiment yourself. Try to play PA (or any kind) in a lobby game. Try to get a crit. Then count if the next attack is also a crit. If this article is wrong, the chance of next attack be a crit should be 15%, because each attack is independent event. If this article is right, you should see the chance of crit is lower than 15% (4%, if I'm not mistaken). The experiment has been repeated many times in history of dota (both 1 and 2), you can look for them.
короче, как в покере, если кто-то понимает о чем я)
@Mandalord
Then I suppose this article is correct in explaining how the Dota 2 mechanic works, however I disagree with the way the 'pseudo' RNGs are coded in Dota 2 using counters. If the purpose is to 'manage luck' as you said, then the best way would be to decrease the variance of the Binomial random numbers simulated from a real RNG. In that way, we can stop this weird trick of predicting when you have the highest chance of crit-ing on PA, and make the critical events independent. There are methods to do decrease variances on random numbers using Antithetic variates, which in my opinion would be a more clever way of coding it.
@Cunnilinguistitian I'm pretty sure he is just saying that in dota 2 the term "pseudo random" doesn't mean what it means in mathematics.
@Cunnilinguistician Ok. We all got your experiences with those random or pseudo-random, how educated and great you are. The point is, we gamers don't care all about those bullshit. We don't care true or pseudo. We only care if the facts can be used to improve our game. I think this article just served its purpose. Just stop making yourself a joke. We all had fun though.
This has been in the game forever and I've never seen a case where someone used it as an advantage. Would be interesting to see if it could actually be done. As far as I can tell there are almost 0 situations where there is both usable time and usable health to sit in the jungle cancelling attacks on creeps to setup an early game gank, as I'm sure this strategy would be near pointless when late game items have been achieved. It's possible though.
@Cunnilinguistician
The article doesn't refer to the generation of pseudorandom numbers in general. In dota 2 the word pseudorandom means that the actual chance to bash/crit whatever is slightly lower than the listed chance and increases with each attack that doesn't crit/bash. This is a system that makes no crit/crit streaks less likely while the average chance to bash/crit is close to the listed one.
That certain random abilities work this way has been confirmed in patch notes when the system was introduced.
Your comment is simply the result of misunderstanding the arguably confusing dota 2 nomenclature.
http://dota2.gamepedia.com/Pseudo-random_distribution
Currently the Enchanted Mango only has a 31.43% win rate. Together, we as a community, have the ability to fix this problem and end this horrific tragedy. Please help. It is our duty as gamers to give this item the respect it is due . KAPPA
@cunnilinguistician I don't know what you guys are talking about exactly, but I'm not sure how much you know about this game: there are two types of random distributions used in it. I have no idea how they work but i do know one is called true random distribution and the other pseudo random distribution.
Best article you guys at dotabuff posted. Learned a lot
Changed top image 3 times XD
hui zalupa chlen jopa ssanina pizda vlagalishe moshonka elda
It can be used very easily since most chances are fairly low and you basicaly have a really high amount of tries to stack them up available... the best example would be PA since you can stack with your autoattacks and proc with the dagger... with every item now too..
Шикарная статья, пойду юзать
I didn't get the first graph. If it's true random, your probability of proc should be a constant 15%...
PHANTOM LANCER´s juxtapose is pure random or is pseudo-random??
i really want to know, i love this hero and it could help me
I tried this with PA and it works like a charm....
Thank you. Wonder Cow
Good Tip
Luck should be reomoved from the game. 'Oh no, I lost 50 MMR because in 2 games I got chain bashed.' This shit skewes matchmaking.
It won't be removed though.