An amoeba that dies, stays, or splits each minute

Question

You start with a single amoeba. Each minute, every amoeba present acts independently: it dies with probability $\tfrac14$ , stays as one amoeba with probability $\tfrac14$ , or splits into two with probability $\tfrac12$ .

What is the expected number of amoebas after three minutes?

Accepted Answer

The population grows in stages, and each stage depends on how many amoebas were around the minute before. The tower rule, conditioning on the previous minute's count, turns this into a clean recursion. Step 1, expected offspring of one amoeba. In one minute a single amoeba turns into a random number of amoebas with mean m = 0 	imes 	frac14 + 1 	imes 	frac14 + 2 	imes 	frac12 = 0 + 0.25 + 1 = 1.25. Step 2, build the recursion with the tower rule. Let Xt be the count after t minutes. Given X{t-1}, the next count is the total offspring of X{t-1} independent amoebas, each producing 1.25 on …

An amoeba that dies, stays, or splits each minute

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