Your mother gifted you a chocolate bunny on your birthday. But this is a magic bunny. On the first night, the bunny either disappears or splits into two identical magic bunnies, with equal probability. The next night, if you have two bunnies, they each (independently) either disappear or split in two. And so it continues, each night any remaining bunnies each independently either disappear or split into two bunnies.

What is the probability that you will eventually be left with no magic bunnies?