So, you want a bad luck prevention system that increases your chance of a legendary drop over time? We already have that.
The game is already pseudo-random, all drops in the game are pseudo-random, all damage in the game is pseudo-random. Computers can't generate true random numbers so all RNG in any game is by definition pseudo-random. Minesweeper is pseudo-random.
Pseudo-random does not mean that the chance is increasing over time, it's the name for computer or mathematically generated random numbers.