30. If a die is tossed 500 times at random, how many times is the probability of getting a 1 (the face with 1 dot facing up) the greatest?
The probability of each point appearing is equal, all being $\frac{1}{6}$.
30. Let $P_{r}$ be the probability that 1 appears exactly $r$ times in 500 tosses. Then, by the formula for the probability of independent repeated trials, we have
$$
\begin{aligned}
& P_{r}=C_{500} \cdot\left(\frac{1}{6}\right)^{r} \cdot\left(1-\frac{1}{6}\right)^{500-r}, \\
\text { Let } & \frac{P_{r+1}}{P_{r}}>1 \\
\Leftrightarrow & \frac{C_{500}^{r+1} \cdot\left(\frac{1}{6}\right)^{r+1} \cdot\left(\frac{5}{6}\right)^{500-r-1}}{C_{500}^{r} \cdot\left(\frac{1}{6}\right)^{r} \cdot\left(\frac{5}{6}\right)^{500-r}}>1 \\
\Leftrightarrow & \frac{500-r}{5(r+1)}>1 \Leftrightarrow r<82.5 \\
\end{aligned}
$$
When $r<82.5$, $P_{r+1}>P_{r}$, $P_{r}$ is increasing; when $r>82.5$, $P_{r+1}<P_{r}$, $P_{r}$ is decreasing, and $\frac{P_{88}}{P_{83}}<1$. Therefore, $P_{83}$ is the largest, i.e., the probability of 1 appearing 83 times is the highest.