This problem occured when I looked into Branching processes, it has to do with the extinction event but I would like to get a more broad understanding of it. So we have given $X_i$ i.i.d. random variables with values in $\mathbb{Z}^+$. Now we also have given a probability for the event $A$ as $P_1(A)=p$. This is in our example is the extinction probability given we start from one individual $X_0=1$. Now it says: Given we start with $k$ individuals, the extinction probability will get $P_k(A)=P(A|Z_0=k)=p^k$.
Well this is somehow clear, when you draw it for example - we start with k individuals. If the probability of on of them to extinct is $p$ and all $k$ of them develop independent from each other, then they will extinct ist the same as if we multiply the amount they are with the probability of one of them going extinct. But how can one formalize that to get to the above $$P_k(A)=P(A|Z_0=k)=p^k$$
Don't we need to show that $$P(A|Z_0=k)=P(A|Z_0=1)\cdot...\cdot P(A|Z_0=1)$$ How can we do so? I would appreciate your help, Keep well!
I think the key insight is that while the random variable $Z_0$ gives you the number of processes to consider, the way each process evolves is independent of $Z_0$. I think it may be more instructive to replace $Z_0$ by another letter, say $N$, to make this clearer.
Say we have $N$ independent and identically distributed branching processes, each starting from a single individual. Let process number $i$ be denoted $(\xi_n^i)_{n \geq 0}$ for $i=1,\ldots,N$ and let $A_i$ denote the event that $(\xi_n^i)_{n \geq 0}$ goes extinct. The probability you are interested in is then $$P(A_1 \cap \cdots \cap A_N | N=k).$$
You can begin by writing $P(A_1 \cap \cdots \cap A_N | N=k) = P(A_1 \cap \cdots A_k | N=k)$. At this point, the event $A_1 \cap \cdots \cap A_k$ is just saying that a fixed, nonrandom number ($k$) of i.i.d. branching processes, each starting from a single individual, will go extinct. How each one of these processes behaves is independent of the random variable $N$. So you can drop the conditioning and write $$P(A_1 \cap \cdots A_k | N=k) = P(A_1 \cap \cdots \cap A_k).$$ From there, it is clear from the i.i.d. assumption that $P(A_1 \cap \cdots \cap A_k) = p^k$.
Does this make sense?