Consider a branching process where $X_n$ represents the number of individuals in generation $n$. Suppose that you know the extinction probability when $X_0 =1$:
$$\alpha=P(extinction|X_0=1)$$
Could you please help me prove that $$P(extinction|X_0=k) =\alpha^k$$
The branching proces under condition $X_0=k$ can be interpreted as $k$ independent branching processes each under condition $X_0=1$.
So there will be extinction under condition $X_0=k$ if there is extinction in these $k$ independent processes under condition $X_0=1$.
Then that event has probability: $$P(\text{extinction}\mid X_0=k)=P(\text{extinction}\mid X_0=1)^k$$
