Final answer:
The question asks for the probability that a branching process with a binomial offspring distribution will eventually die out. This involves using generating functions to solve for the extinction probability.
Step-by-step explanation:
The subject question revolves around a branching process, which is a type of random process that serves as a mathematical model for populations.
The branching process described has an initial population size (X0) of 1, and the offspring distribution is given by a binomial distribution with parameters 3 and 1/2, denoted as V ~ Bin(3, 1/2).
The probability that the process will eventually die out is computed using the theory of generating functions for branching processes.
To calculate the extinction probability, one would set up the generating function for the given binomial distribution and solve for the probability that the process converges to zero.
This involves finding the smallest non-negative solution to the equation G(s) = s, where G(s) is the generating function for the binomial distribution.