Geodesic
A prisoner problem variation
Journal of integer sequences, Tome 18 (2015) no. 2
Consider a fair $n$-sided die with faces numbered 1 to $n$. Several different methods are used to compute the probability that every face has come up at least once when face $n$ appears for the $k^{th}$ time. The results lead to a number of summation identities. The probabilities are related to several sequences in Sloane's On-Line Encyclopedia of Integer Sequences.
Classification : 11B99, 60C05
Keywords: occupancy problem, waiting time, inclusion-exclusion counting, Stirling number of the second kind, Markov chain, binomial identity, binomial sum, multinomial sum, binomial transformation 
@article{JIS_2015__18_2_a1,
     author = {Metzger,  Jerry and Richards,  Thomas},
     title = {A prisoner problem variation},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {2},
     zbl = {1378.60074},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_2_a1/}
}
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Metzger,  Jerry; Richards,  Thomas. A prisoner problem variation. Journal of integer sequences, Tome 18 (2015) no. 2. http://geodesic.mathdoc.fr/item/JIS_2015__18_2_a1/