Example 12. The weight of the caught fish follows a normal distribution with parameters $a=375$ g, $\sigma=25$ g. Find the probability that the weight of one fish will be: a) from 300 to 425 g; b) no more than 450 g; c) more than 300 g.
Solution. We use formula (3.5.2), setting $a=375$ g, $\sigma=25$ g.

In the first case $\alpha=300, \beta=425$, therefore

$$
\begin{gathered}
P(300< X < 425) = \Phi\left(\frac{425-375}{25}\right) - \Phi\left(\frac{300-375}{25}\right) = \\
= \Phi(2) - \Phi(-3) = \Phi(2) + \Phi(3) = 0.97725 + 0.49865 = 0.97725 + 0.49865 = 0.9759
\end{gathered}
$$

For the second case, where we need to find $P(X>300)$, we get

$$
\begin{gathered}
P(X>300)=P(300<X<+\infty)=\Phi\left(\frac{+\alpha-375}{25}\right)-\Phi\left(\frac{300-375}{25}\right)= \\
=\Phi(+\infty)-\Phi(-3)=\Phi(+\infty)+\Phi(3)=0.5+0.49865=0.9987
\end{gathered}
$$