Example 20. The diameter of a part manufactured in the workshop is a random variable distributed according to the normal law with parameters $a=4.5$ cm and $\sigma=0.05$ cm. Find the probability that the diameter size of a randomly taken part differs from the mathematical expectation by no more than $1$ mm.
Solution. Applying formula (3.5.4) with $\delta=0.1$ cm, we obtain the required probability.

$$
P(|X-4.5| \leq 0.1)=2 \Phi\left(\frac{0.1}{0.05}\right)=2 \Phi(2)=2 \cdot 0.4772=0.9544
$$