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?
(In reply to
A slightly simpler solution by Steve Herman)
It seems I misread the problem, but had no effect on the solution. So I'll compare the two.
In my problem the expected amount after is (0+1+2)/3=1, and for the problem as written it is (0+2)/2=1. In both cases the population of bunnies wants to stay the same.
Also the number of bunnies can be thought of like a random walk: disappear is step 1 to the left and splitting in 2 is step 1 to the right. The walk is expected to be centered on the original population but can wander to some arbitrary point, and zero is one of those points.
So now I will entertain the idea of the bunnies tripling. In my case the expected amount is (0+1+3)/3=1.333 or as written (0+3)/2=1.5. Here the population wants to grow so we will have some positive probability that the bunnies will not go extinct
Using the random walk analogy now the walk is weighted to the right. We no longer expect the walk to be centered at the original population, instead it will trend to the right while zero is to the left.
re: A slightly simpler solution
