Random Distribution

Pseudo-random distribution (often shortened to PRD) in DotA refers to a statistical mechanic of how certain probability-based items and abilities work. In true random distribution, every "roll" operates independently, but in PRD, the effect's chance increases every time it does not happen. This results in the effects occurring more consistently. In general, PRD is applied to the following types of abilities: Critical Strike, Bash, Damage Block, Chain Lightning, Maim and Evasion.

The Uniform random distribution or True random distribution describes the probability of random event that are not pseudo-random. In general this applies to Blind as well as most non-damage or non-hero-unit related abilities.

Summary
In general randomness is rejected from competitive gaming since it can decide over loss and defeat without reflecting the actual skill level. Nonetheless there are a couple of random events within DotA 2, that keep the game less predictable and static. To limit the influence of huge streaks, the pseudo-random distribution replaced the true random distribution, where it made sense for the developers.

Summary
The probability of an effect to occur (or proc) on the Nth test since the last successful proc is given by P(N) = C N. For each instance which could trigger the effect but does not, the PRD augments the probability of the effect happening for the next instance by a constant C. This constant, which is also the initial probability, is lower than the listed probability of the effect it is shadowing. Once the effect occurs, the counter is reset.


 * Example:


 * 's has a 25% chance to stun the target. On the first attack, however, it only has an ~8.5% probability to bash. Each subsequent attack without a bash increases the probability by ~8.5%. So on the second attack, the chance is ~17%, on the third it is ~25.5%, etc. After a bash occurs, the probability resets to ~8.5% for the next attack. These probabilities average out so that, over a moderate period of time, Bash of the Deep procs nearly 25% of the time.

Effects based on PRD rarely proc many times in a row, or go a long time without happening. This makes the game 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.

Aside from damage block items, the probabilities listed in the game are theoretical percentages, which are slightly higher than the actual probabilities. In the following table, P(T) is the theoretical probability while P(A) is the actual average. C is the PRD constant. Max N is the minimum number of attacks that would result in C N becoming greater than 1 (i.e. guaranteed proc). Average N is the expected value of N: the sum of the products of N and probabilities. SD is the standard deviation of N, a measure of how spread the data is, using the population formula. The lower the deviation, the more consistent the procs are. For comparison, SDt is the standard deviation of N in true random distribution, using the sample formula from N = 1 to N = 264. Apparently these are higher than those of PRD, hence not as consistent.

Flat distribution
Some mechanics roll a random value between 0 and 1 and scale it along the minimum / maximum gradient. This includes


 * Physical right click damage
 * Including heroes, summons, creep and tower damage.
 * Excluding some wards.
 * Last hit gold bounty.
 * Including enemy and neutral creeps, most summons, towers, barracks and filler buildings.
 * Excluding heroes, creep-heroes and denies.
 * Roshan respawn timer (8 to 11 minutes).

Discrete distribution
A couple of random events have discrete states that have similar probability.
 * Powerup Runes with a discrete clamping over the 6 types and 2 locations.
 * Neutral camps can spawn different creeps for each of the small, medium, hard and ancient camps.

Blind
Since 6.87, Evasion uses pseudo random distribution. It was not mentioned whether Blind and the 25% uphill miss chance are affected by this change. So far no testing has been done.