Random Distribution

The Pseudo-random distribution (hereafter shortened to PRD) in Dota 2 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, Daedalus has a 25% probability to critical strike for 2.5x damage. On the first attack, however, it will only have an ~8.5% probability to crit; this is its hidden PRD constant. Each subsequent attack without a crit 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 crit procs, the probability resets to 8.5% for the next attack. These probabilities average out so that, over a moderate period of time, Daedalus will crit 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 Daedalus, a critical strike 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 Crystalys 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/items that use PRD

 * Any source of damage block (such as Vanguard)
 * Any source of critical strike, such as Daedalus
 * Any source of Bash, such as Slardar's Bash (except Skull Basher, which does not use PRD)
 * Maim from Sange or Sange and Yasha and similar abilities
 * Maelstrom and Mjollnir

Effects with >25% stated probability
There is one important exception to note. Due to legacy reasons (the Warcraft III engine the original DotA used), PRD for items that use it and are listed as having a greater than 25% chance to proc, actually have a lower probability. Vanguard, for instance, which is listed as having a 70% chance, actually has closer to 60% chance to block damage. The multiplier for each instance is ~42%, so it will always proc after 3 hits and usually after 2. Note that anything listed as having a 100% chance will always work (only relevant for Poor Man's Shield. )

PRD abilities over 25%

 * Stout Shield: listed as 60%, actually ~53%
 * Poor Man's Shield: listed as 60%, actually 53% (100% for hero-based damage)
 * Vanguard: listed as 70%, actually ~60%