Example 10. How many times should a die be thrown so that the most probable number of times a two appears is 32?
Solution. In this case, $k_{0}=32, p=\frac{1}{6}, q=\frac{5}{6}$. The inequalities $n p-q \leq k_{0} \leq n p+p$ become

$$
n \cdot \frac{1}{6}-\frac{5}{6} \leq 32 \leq n \cdot \frac{1}{6}+\frac{1}{6}
$$

or

$$
n \cdot \frac{1}{6}-\frac{5}{6} \leq 32 ; \quad n \cdot \frac{1}{6}+\frac{1}{6} \geq 32
$$

From the first inequality, it follows that $n \leq 197$, and from the second, that $n \geq 191$. Thus, it is necessary to conduct between 191 and 197 independent trials.