Random Distribution

The Pseudo-random distribution (often shortened to PRD) in DotA refers to the statistical mechanics 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.

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 doesn't, 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. For example, Slardar's Bash has a 25% to Stun the target. On the first attack, however, it will only have 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 will proc 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 is higher than the actual probability. 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.