Polymorphic Bacteria
Problem 666
If a population starts with a single bacterium of type , then it can be shown that there is a 0.07243802 probability that the population will eventually die out, and a 0.92756198 probability that the population will last forever. These probabilities are given rounded to 8 decimal places.
Now consider another species of bacteria, (where and are positive integers), which occurs in different types for . The rules governing this species' lifecycle involve the sequence defined by:
Every minute, for each , each bacterium of type will independently choose an integer uniformly at random in the range . What it then does depends on :
In fact, our original species was none other than , with and .
Let be the probability that a population of species , starting with a single bacterium of type , will eventually die out. So . You are also given that and , all rounded to 8 decimal places.
Find , and give your answer rounded to 8 decimal places.