Consider a branching process with offspring distribution {βk}k≥0 and assume 0 < β0 < 1. Suppose the evolution is altered as follows. Start with X0 = 1, a single individual. If the next generation is not empty (that is, if X1 > 0), they invite an immigrant to join them (that is, X1 is replaced by X1 + 1). After that the evolution continues according to the usual rule. In particular, no further immigration takes place. Find the extinction probability ˆπ of this new process, expressed in terms of β0 and the original extinction probability π. For which values of π do we have ˆπ < π, in other words, the immigrant strictly improves the chances of survival? Hint: Condition on the number of offspring of the first individual.