(11) (12 points) There are $n$ keys that look the same, among which only one can unlock the door. Suppose the keys are drawn independently and with equal probability, and each key is not returned after being tried, find the distribution of the number of trials $\varepsilon$ and the mathematical expectation $E \varepsilon$.

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The translation retains the original text's formatting and structure.
(11) The distribution of $\varepsilon$ is
\begin{tabular}{|c|c|c|c|c|}
\hline$\varepsilon$ & 1 & 2 & $\cdots$ & $n$ \\
\hline$P$ & $\frac{1}{n}$ & $\frac{1}{n}$ & $\cdots$ & $\frac{1}{n}$ \\
\hline
\end{tabular}
$$
E_{\varepsilon}=1 \cdot \frac{1}{n}+2 \cdot \frac{1}{n}+\cdots+n \cdot \frac{1}{n}=\frac{n+1}{2} .
$$