8 A certain station has exactly one bus arriving during $8: 00-9: 00, 9: 00-10: 00$ every day, but the arrival time is random, and the arrival times of the two buses are independent, with the following distribution:
\begin{tabular}{|c|c|c|c|}
\hline Arrival time & \begin{tabular}{c}
$8: 10$ \\
$9: 10$
\end{tabular} & \begin{tabular}{l}
$8: 30$ \\
$9: 30$
\end{tabular} & \begin{tabular}{l}
$8: 50$ \\
$9: 50$
\end{tabular} \\
\hline Probability & $\frac{1}{6}$ & $\frac{1}{2}$ & $\frac{1}{3}$ \\
\hline
\end{tabular}

A passenger arrives at the station at $8: 20$, then the expected waiting time for the bus is $\qquad$ (to the nearest minute).
827 Prompt: The distribution of passengers' waiting time is as follows:
\begin{tabular}{|c|c|c|c|c|c|}
\hline Waiting Time (min) & 10 & 30 & 50 & 70 & 90 \\
\hline Probability & $\frac{1}{2}$ & $\frac{1}{3}$ & $\frac{1}{6} \times \frac{1}{6}$ & $\frac{1}{2} \times \frac{1}{6}$ & $\frac{1}{3} \times \frac{1}{6}$ \\
\hline
\end{tabular}

The mathematical expectation of the waiting time is
$$
10 \times \frac{1}{2} + 30 \times \frac{1}{3} + 50 \times \frac{1}{36} + 70 \times \frac{1}{12} + 90 \times \frac{1}{18} \doteq 27 .
$$