Random Distribution

The Pseudo-random distribution (hereafter shortened to PRD) in DotA refers to the statistical mechanics of how certain probability-based items and abilities work. Unlike a true random distribution, where the chance of an effect activating is unaffected by previous attempts to trigger it, hidden iterating counters are used to prevent long lucky or unlucky streaks.

Summary
Essentially, for each instance which could trigger the effect but doesn't, the PRD augments the probability of the effect happening for the next instance by a certain constant. This constant (which is also the initial probability) is usually quite low compared to the stated probability of the effect it is shadowing. This probability counter resets every time an instance of the effect occurs. Over a moderately large period of time, the expected probability for each instance is almost exactly the listed probability (but see the note below), and since the probability increases steadily for each time the effect doesn't happen (and resets when it does happen), the effect occurs more consistently.

For example, Slardar's Bash has a 25% probability to Stun the target. On the first attack, however, it will only have an ~8.5% probability to bash; this is its hidden PRD constant. Each subsequent attack without a bash increases this probability by 8.5%. So on the second attack, the chance is 17%, on the third it is 25.5%, etc. After a bash procs, the probability resets to 8.5% for the next attack. These probabilities average out so that, over a moderate period of time, Bash will proc very nearly 25% of the time.

The important gameplay and balance effect of PRD is that effects based on it rarely occur many times in a row, or go a long time without happening. In the case of Bash, a bash will occur very nearly every four attacks, will rarely occur twice in a row, and will always happen within 11 attacks. This makes the game far 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.

List of abilities that use PRD

 * Hero Abilities that grant critical strike, namely Mortred's Coup de Grace, Brewmaster's Drunken Brawler, Juggernaut's Blade Dance, Chaos Knight's Chaos Strike, Skeleton King's Critical Strike and Lycanthrope's Critical Strike.
 * Most Hero Abilities that grant Bash or Mini-Bash, namely Slardar's Bash, Faceless Void's Time Lock and Sniper's Headshot. The only exception is Spirit Breaker's Greater Bash.

Note: In DotA 1, Items like Stout Shield or Crystalys used PRD as well. This feature has not been ported to DotA 2, however.