Credit for this problem goes to The Riddler. Thanks!

Survival of the species

From “The Riddler”:

From Thierry Zell, a “multiplication” problem he recalled from his grad school days:

At the beginning of time, 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?