Q&A

# Intuitively, why would organisms — that after one minute, will either die, split into two, or stay the same, with equal probability — all die ultimately?

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I have no questions on the solution or the algebra, but even after re-reading the solution, I still can't fathom or intuit why $P(D) = 1$ from the problem statement. Even now, I couldn't have divined or foretold that $P(D) = 1$!

The strategy of first-step analysis works here because the problem is self-similar in nature: when Bobo continues as a single amoeba or splits into two, we end up with another version or another two versions of our original problem. Conditioning on the first step allows us to express P(D) in terms of itself. □

Blitzstein. Introduction to Probability (2019 2 ed). p 72.

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This can be recast as a random walk on a line. Let $n_t$ be the number of amoebae after $t$ events, and process the events in any order which makes sense. (It may help to think of this as serialising a parallel process on a single-core CPU). For example, you could choose to number the the amoebae by the $t$ at which they are created and always process the lowest-numbered living amoeba. Now at each event $t$ we can decrement the number of amoebae, increment it, or do nothing.

To see informally that a random walk on an infinite line with equiprobable steps of ${-1, 0, 1}$ eventually hits zero, suppose that at event $t$ it hasn't yet hit zero, i.e. $n_t > 0$. Then by symmetry it's 50/50 whether it will reach zero or $2n_t$ first. Avoiding eventually hitting zero is like tossing a fair coin infinitely many times without ever getting tails.

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