Example 8. The probability of an event occurring in each of 100 independent trials is constant and equal to $p=0.8$. Find the probability that the event will occur: a) not less than 70 and not more than 85 times; b) not less than 70 times; c) not more than 69 times.
Solution. We will use formula (4.3.7) in cases a) and b).

a) According to the condition, $n=100, p=0.8, q=0.2, k_{1}=70, k_{2}=85$. Using formulas (4.3.6), we calculate $x_{1}$ and $x_{2}$:

$$
x_{1}=\frac{70-100 \cdot 0.8}{\sqrt{100 \cdot 0.8 \cdot 0.2}}=\frac{70-80}{\sqrt{16}}=-\frac{10}{4}=-2.5
$$

$$
x_{2}=\frac{85-80}{4}=\frac{5}{4}=1.25
$$

Using the table of values of the Laplace function (see appendix), and considering the oddness of this function, we find

$$
\begin{gathered}
\Phi\left(x_{1}\right)=\Phi(-2.5)=-\Phi(2.5)=-0.4938 \\
\Phi\left(x_{2}\right)=\Phi(1.25)=0.3944
\end{gathered}
$$

According to formula (4.3.7), we get

$$
m_{100}(70 ; 85)=\Phi(1.25)-\Phi(-2.5)=0.3944+0.4938=0.8882
$$

b) The requirement that the event occurs at least 70 times means that the number of occurrences of the event can be 70, 71, 72, ..., or 100. Therefore, in this case, we should take $k_{1}=70$, $k_{2}=100$, then

$$
\begin{gathered}
x_{1}=\frac{70-100 \cdot 0.8}{\sqrt{100 \cdot 0.8 \cdot 0.2}}=\frac{70-80}{\sqrt{16}}=-\frac{10}{4}=-2.5 ; \\
x_{2}=\frac{100-100 \cdot 0.8}{\sqrt{100 \cdot 0.8 \cdot 0.2}}=\frac{20}{4}=5 .
\end{gathered}
$$

Using the table of values of $\Phi(x)$, we find

$$
\Phi(-2.5)=-\Phi(2.5)=-0.4938 ; \Phi(5)=0.5
$$

Based on formula (4.3.7), we get

$$
m_{100}(70 ; 100)=\Phi(5)-\Phi(-2.5)=0.5+0.4938=0.9938
$$

c) The events "A occurs at least 70 times" and "A occurs at most 69 times" are complementary, therefore

$$
P_{100}(0 ; 69)=1-m_{100}(70 ; 100)=1-0.9938=0.0062
$$