I am having trouble with this task, I feel like I'm doing something wrong but I dont know what:
Let $X_1, X_2 \cdots X_n $ $ n \geq 2 $ be independent Bernoulli Experiments. With a not specified "hitrate" of $p$.
Further let $Y = \sum^{n-1}_{i=0} X_i $ and $Z = \sum^n_{i=0} X_i $.
Also let $k \in \mathbb{N} \ \ \land k \neq 0 \land k \neq n $
Now the actual question:
Express the event $\{ Z =k\} \cap \{Y = k-1\} $ and the event $\{ Z=k \} \cap \{Y =k \} $ with the random variables $ Y $ and $X_n$ without using $Z$ or $X_1, \cdots, X_{n-1} $
IMO the first one is:
$$ P(Y=k-1) \cdot P(X_n = x)$$
and the second one should be:
$$ P(Y=k-1) / P(X_n = x) $$
Can someone please look over my reasoning and correct me if I'm wrong. This isnt homework and only preparation for an exam.