The Random Generator is BPS's name for the algorithm used to generate the sequence of tetrominoes in Tetris brand games that follow the Tetris Guideline.
Random Generator generates a sequence of all seven one-sided tetrominoes (I, J, L, O, S, T, Z) permuted randomly, as if they were drawn from a bag. Then it deals all seven tetrominoes to the piece sequence before generating another bag. There are 7!, or 5, 040, permutations of seven elements, and it is believed that assigns a nearly equal probability to each of these, making it much less likely that the player will get an obscenely long run without a desired tetromino. It can produce a maximum of 12 tetrominoes between one I and the next I, and a run of S and Z tetrominoes is limited to a maximum of 4. Exception: In Random Generator as implemented in Tetris The Grand Master Ace, the first piece of the first bag is always I, J, L, or T, just as in the traditional TGM randomizer.
While the number of tetrominoes in a single bag is usually 7, some games use a different number. The public beta of Tetris Online (Japan) used an 8-bag randomizer for the player. Not all guideline-compliant games use the Random Generator in all modes. and, for instance, use a different randomizer in their Square modes, and TGM3 uses the TGM randomizer even when the game is set to "World" mode.
There are two "snake" tetrominoes, called S and Z. As only two snakes will be in a given bag, a sequence of more than two snakes must cross the "seam" between bags. The probability of the next two bags having a sequence of four consecutive snakes, the maximum possible, is 1/(7*6*7*6) for SZSZ and likewise for SZZS, ZSSZ, and ZSZS, for a total of 1/441. But the probability of these being your three next pieces are 1/441 times the probability of being at position 6 in a bag, so the probability of the next four pieces being SZSZ are 1 in 3087.
Define a "2|1 combo" as chosen sixth and seventh pieces in one bag and first piece in next bag, and a "1|2 combo" as chosen seventh piece in one bag and first and second pieces in next bag. Define a "snake" as the S tetromino or the Z tetromino.
The probability of any 2|1 combo (e.g. SZ|Z) is 1/(7*6*7) = 1/294. There are four different 2|1 combos containing all snakes (SZ|Z, SZ|S, ZS|Z, and ZS|S), so the probability of getting a 3-snake 2|1 in your next two bags is 4/294. But the probability of being at the sixth piece in a bag, where your next three pieces are a 2|1, is 1/7, making the probability of being at a three-snake 2|1 equal to 4/(294*7) = 2/1029. By symmetry, the 1|2 probabilities are exactly the same: 2/1029. So for Random Generator, this makes a 1 in 257 chance of your next three tetrominoes being snakes.