perplexus.info :: Probability : Magic Bunny Power

Magic Bunny Power (Posted on 2024-08-04)

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?

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

| Comment 2 of 7 |

Let p(n) represent the probability of a group of n bunnies eventually dying out. p(0)=1 and we want to find p(1).

We start with 1 bunny which can become 0, 1, or 2 bunnies; equally likely. Then p(1) = 1/3*p(0) + 1/3*p(1) + 1/3*p(2).

If we have n bunnies at some point, they are effectively n independent instances of one bunny. Then the probability of all them dying out is p(1) multiplied out for n bunnies: p(n) = p(1)^n.

Combine everything to make the equation p(1) = 1/3 + (1/3)*p(1) + (1/3)*p(1)^2. This can be rearranged and simplified to (p(1) - 1)^2 = 0. The only root is p(1)=1, so we conclude the bunnies will eventually die out regardless of how many there are at one time.

Posted by Brian Smith on 2024-08-04 14:44:17

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