# [R], 25 bytes

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    c(sample(99,rexp(1),T),0)

[Try it online!][TIO-kkh7l6ia]

[R]: https://www.r-project.org/
[TIO-kkh7l6ia]: https://tio.run/##dcwxCsAgDADA3Ve4mYCDjvqOfiBICoLaoBbs620/0PWG63snGFSlMIRgOy8Bj/ZA63AnmmBUZ2Ga2junZ648olIG1SCR8oCPn9vzbmnmq8HC3w33Cw "R – Try It Online"

p(0) at each iteration is (e-1)/e.  
p(each other number) at each iteration is (1/e)*(1/99).

Obviously this choice of random distribution gives a rather unsatisfying-looking output (since most of the runs are rather short).  So [this link](https://tio.run/##fcwxCsAgDADA3Ve4NYFQdNR39AMiKQhqg1qwr7d9Qdcbrq0VoYcimcE5ajwFLO3GIh1IBlcMAzbVWDgMbY3RIxXuXqkNVQ8i@QHrP6fzrnGkq8LE3xHXCw) uses the same approach, but changes p(0) to roughly 0.01 to illustrate some longer runs...