Example 15. A die is rolled 80 times. Find, with a probability of 0.99, the boundaries within which the number $m$ of fives will be contained.
Solution. We will use formula (4.3.9). First, we will find the number $\varepsilon>0$, and then the boundaries within which the number $m$ is confined. According to the condition $n=80$; in this case $p=1 / 6, q=5 / 6$. In accordance with the condition

$$
\begin{gathered}
2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)=0.99, \quad 2 \Phi\left(\varepsilon \sqrt{\frac{80}{1 / 6 \cdot 5 / 6}}\right)=0.99 \\
2 \Phi(24 \varepsilon)=0.99, \quad \Phi(24 \varepsilon)=0.495
\end{gathered}
$$

From the table of values of the Laplace function (see appendix), we find that $\Phi(2.58)=0.495$; hence, $24 \varepsilon=2.58$, from which $\varepsilon=0.11$. From the inequality

$$
\left|\frac{m}{80}-\frac{1}{6}\right| \leq 0.11
$$

we determine that $0.06 \leq m / 80 \leq 0.28$, or $4.8 \leq m \leq 22.4$.

Since $m$ is an integer, then $5 \leq m \leq 22$.