Paper 1, Section I, F

Probability | Part IA, 2020

A robot factory begins with a single generation-0 robot. Each generation- nn robot independently builds some number of generation- (n+1)(n+1) robots before breaking down. The number of generation- (n+1)(n+1) robots built by a generation- nn robot is 0,1,20,1,2 or 3 with probabilities 112,12,13\frac{1}{12}, \frac{1}{2}, \frac{1}{3} and 112\frac{1}{12} respectively. Find the expectation of the total number of generation- nn 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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