another Galton-Watson riddle

The riddle on the Riddler this week is definitely a classic, since it rephrases the standard Galton-Watson branching process (which should have been called Bienaymé‘s process, as he established the relation before Watson, while the jack-of-all-trades Francis Galton only posed the question):

At the beginning, there is a single microorganism. Each day, every member of this species either splits into two copies of itself or dies. If the probability of multiplication is p, what are the chances that this species goes extinct?

As is easily seen from the moment generating function, the species goes instinct if p≤½. Actually, I always found it intriguing [intuitively] that the value ½ is included in the exclusion range!

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