Paper 1, Section I, F

A robot factory begins with a single generation-0 robot. Each generation- $n$ robot independently builds some number of generation- $(n+1)$ robots before breaking down. The number of generation- $(n+1)$ robots built by a generation- $n$ robot is $0,1,2$ or 3 with probabilities $\frac{1}{12}, \frac{1}{2}, \frac{1}{3}$ and $\frac{1}{12}$ respectively. Find the expectation of the total number of generation- $n$ robots produced by the factory. What is the probability that the factory continues producing robots forever?

[Standard results about branching processes may be used without proof as long as they are carefully stated.]

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