Example 7. The probability of manufacturing a first-class part on a given machine is 0.8. Find the probability that among 100 randomly taken parts, 75 will be of the first class.
Solution. We will find the required probability using formula (4.3.2). According to the condition, $n=100, k=75, p=0.8$; hence $q=1-p=0.2$. According to formula (4.3.3), we calculate

$$
x=\frac{k-n p}{\sqrt{n p q}}=\frac{75-100 \cdot 0.8}{\sqrt{100 \cdot 0.8 \cdot 0.2}}=\frac{75-80}{\sqrt{16}}=\frac{-5}{4}=-1.25
$$

from the table, we find $\varphi(-1.25)=\varphi(1.25)=0.1826$.

Therefore,

$$
P_{100}(75)=\frac{1}{4} \varphi(-1.25)=\frac{0.1826}{4}=0.04565
$$