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?
What if the magic bunny either disappears or splits into three identical magic bunnies?
Again, let p = probability that you will eventually be left with no magic bunnies.
Then, p = (1/2)*1 + (1/2)*p^3
p^3 - 2p +1 = 0
Roots are 1, and (-1+/- sqrt(5)/2
What can we make of this?
The negative root is clearly not our desired probability. Somewhat less clearly, neither is p = 1. On average, a magic bunny is expected to turn into 1.5 magic bunnies every night. This is not a process that is guaranteed to eventually result in no bunnies.
No, the answer is (-1+sqrt(5))/ 2 = Phi - 1 = 1/Phi = approx .618034. Phi is the Golden Ratio!
61.8% of the time we are left with no bunnies.
38.2% of the time we wind up, I think, with an infinite number of bunnies.
Edited on August 6, 2024, 7:49 am
The Golden Ratio appears
