Solution. From the condition of the problem, it follows that $n=1000, \varepsilon=0.01$, $p=0.03, q=1-p=0.97$.

In accordance with formula (4.2.5.), we obtain

$$
P\left(\left|\frac{m}{n}-p\right| \leq 0.01\right) \geq 1-\frac{p q}{n \varepsilon^{2}}=1-\frac{0.03 \cdot 0.97}{10000 \cdot(0.01)^{2}}=1-\frac{0.0291}{0.1}=0.709
$$

Thus, the required probability $P \geq 0.709$.
Example 2. In the stamping of plastic plates, the defect rate is $3 \%$. Find the probability that when checking a batch of 1000 plates, the deviation from the established defect rate will be less than $1 \%$.