Bacteria reproduce by cell division. In a unit of time, a bacterium will either die (with, $p=\frac{1}{4}$), stay the same (with, $p=\frac{1}{4}$), or split into 2 parts (with, $p=\frac{1}{2}$). The population starts with 100 bacteria at time $n = 0$. What is the extinction probability for this population?
I have calculated $E(a^y)=a$ and, got $a= -\frac{1}{2}$ and $a=1$.
Which should I accept as I know you are meant to pick the smallest answer but it does not make sense how a probability could be negative?
