Example 6.18. When a technological process is established, a factory produces on average $70 \%$ of first-grade products. What is the probability that in a batch of 1000 items, the number of first-grade items is between 652 and 760?
Solution. The number of independent trials $n=1000$ and the probability of the event occurring in a single trial $p=0.7$ are known. It is required to find the probability that the number of occurrences of the event is between $a=652$ and $b=760$. The required probability is found using formula (5)

$$
P(a<m<b) \approx \frac{1}{2}[\Phi(\beta)-\Phi(\alpha)] .
$$

The calculation of $\alpha$ and $\beta$ is carried out according to the following scheme:

$$
\begin{aligned}
& n p=0.7 \cdot 1000=700 \\
& n p q=700 \cdot(1-0.7)=700 \cdot 0.3=210 \\
& \sqrt{n p q}=\sqrt{210}=14.491 \\
& a-n p=652-700=-48 \\
& \alpha=\frac{a-n p}{\sqrt{n p q}}=\frac{-48}{14.491}=-3.31 \\
& b-n p=760-700=60 \\
& \beta=\frac{b-n p}{\sqrt{n p q}}=\frac{60}{14.491}=4.14 \\
& -\Phi(\beta)=\Phi(4.14)=0.99997 \\
& \frac{\Phi(\alpha)=\Phi(-3.31)=-0.99813}{\Phi(\beta)-\Phi(\alpha)=1.99810}
\end{aligned}
$$

Therefore, the required probability is approximately equal to:

$$
\frac{\Phi(\beta)-\Phi(\alpha)}{2}=\frac{1.99810}{2}=0.99905
$$