331. The measurement of the shaft diameter is being performed without systematic (one-sided) errors. Random measurement errors $X$ follow a normal distribution with a root mean square deviation $\sigma=10$ mm. Find the probability that the measurement will be performed with an error not exceeding $15 \mathrm{mm}$ in absolute value.
Solution. The mathematical expectation of random errors is zero, so the formula $P(|X|<\delta)=2 \Phi(\delta / \sigma)$ can be applied. Setting $\delta=15, \sigma=10$, we find $P(|X|<15)=2 \Phi(1.5)$. From the table in Appendix 2, we find: $\Phi(1.5)=0.4332$. The required probability

$$
P(|X|<15)=2 \cdot 0.4332=0.8664 .
$$