Given a real random number generator with uniform distribution of values between $1$ and $N$. The random number series ends if all values between $1$ and $N$ have been picked at least once.
How high is the chance that any of those numbers have been picked $t$ or more times? $(t>1)$
Solving attempt:
(approximation)
How many numbers needed to get all values? $=T$
->Coupon collector's problem
$$E(T) = N \cdot H_N = N \log N + \gamma N + \frac{1}{2} + O(1/N)$$ $$\gamma \approx 0.5772156649$$
Number of values which are duplicates $D$: $$D=E(T)-N$$
Chance of $t$ or more repetitions of one number (binomial distribution):
$$p_1(>=t) = \sum_{k=t}^D \binom{D}{k}\frac{1}{N^k}(1-\frac{1}{N})^{D-k}$$
...but this is only for one number. Any idea how to compute the probability of all numbers?
Or some alternative way of solving this?
Edit: Findings so far
from antkam: $$T-1 > (t-1)\cdot (N-1)$$ with this also $$t >= \frac{T-1}{N-1}$$