n-faced dice thrown n times

Question

We have a n-faced dice that we throw n times.

p(n) is the probability to find a particular number / face.

p(1) = 1

p(2) = 0.75

p(3) ~= 0.703704

For a n that reaches infinity, what is p(n) ?

(To what number does the trend converge ?)

It might be 2*Pi but I'm not too sure

Answer 1

Probability of finding face $j$ in $n$ throws is $$P\bigg(\bigcup_{k=1}^n\{X_k=j\}\bigg)=1-\prod_{k=1}^nP(X_k \neq j)=1-\bigg(\frac{n-1}{n}\bigg)^n=1-\bigg(1-\frac{1}{n}\bigg)^n\to 1-e^{-1}$$