The pub Gosset’s Paradise serves $N ≥ 1$ different brands of beer. Each time a drink is ordered, one of the brands is served uniformly at random. John wants to sample each brand at least once. Let $X_n ∈ {1,...,N}$ denote the brand that was served at the $nth$ order. Let $E_n ⊂ {1,...,N}$ be the random set of different brands that John has been served after his $nth$ order.
Let $T_r$ $:=$ $inf${$n$ $≥$ $1$ : |$E_n$| = $r$} denote the first time when John has sampled $r ∈ {1,...,N}$ different brands.
Finally, set $D_r$ $:=$ $T_r$ $−$ $T_{r−1}$ and $T_0$ $:=$ $0$.
Let $1$ $≤$ $r$ $<$ $N$ and $d_1$,...,$d_r$ $∈$ {${1,2,...}$}
The question is as follows: Let $d$ $∈$ {${1,2,...}$} and $I$ $⊂$ {${1,...,N}$} with $|I|$ $=$ $r$. Show that for every $i$ $\notin$ $I$ we have
$\mathbb P(A|B)$ $=$ $(\frac{r}{N})^{d-1}$ $*$ $\frac{1}{N}$,
where we take
$A = \{T_{r + 1} = n + d\} \cap\,\{ X_{n + d} = i \} $ and $B=\{ D_j = d_j \text{ for $1\le j \le r$}\} \cap \{\Xi_n = I\}$
and $n$ $:=$ $d_1$ $+$ $···$ $+$ $d_r$.
After that, I have to show that $D_{r+1}$ is geometrically distributed with parameter $1$ $-$ $\frac{r}{N}$.
I know this question looks like the Coupon's collector problem, but I have a lot of trouble finding an answer to this question due to all the "variables" and information.