In general, a paralyzed Pokémon runs a 25% risk of losing their turn due to full paralysis. In addition, the afflicted Pokémon's Speed decreases to 25%.
This is maths I like to call consecutive probability. I explain it here, but it is basically the chance of the same thing happening over and over again.
Say you use /dice 4 on PS!, which picks a random number between one and four. With each time you use the command, the chance of picking the same number decreases. Let's let 4 be the instance where you don't attack, and 1-3 be where paralysis doesn't effect you. Since there are 7 instances of this, we'll do the calculation seven times:
/dice 4: 1 2 3 4 (1/4 chance)
/dice 4: 1 2 3 4 (1/4 x 1/4 chance)
/dice 4: 1 2 3 4 (1/4 x 1/4 x 1/4 chance)
/dice 4: 1 2 3 4 (1/4 x 1/4 x 1/4 x 1/4 chance)
/dice 4: 1 2 3 4 (1/4 x 1/4 x 1/4 x 1/4 x 1/4 chance)
/dice 4: 1 2 3 4 (1/4 x 1/4 x 1/4 x 1/4 x 1/4 x 1/4 chance)
/dice 4: 1 2 3 4 (1/4 x 1/4 x 1/4 x 1/4 x 1/4 x 1/4 x 1/4 chance)
So, since you want the consecutive chance, you multiply the chance of picking said number (4) by itself for the number of instances there are. In this case, 7. You multiply it all together and you get a:
#0.006103515625% chance of this happening
