1586. The share of premium quality items in this enterprise is $31 \%$. What is the most probable number of premium quality items in a randomly selected batch of 75 items?
Solution. It is known that $p=0.31, n=75$, and $q=1-p=$ $=0.69$. We form the double inequality (1):

$$
\begin{gathered}
75 \cdot 0.31 - 0.69 \leqslant m_{0} \leqslant 75 \cdot 0.31 + 0.31 ; \\
22.56 \leqslant m_{0} \leqslant 23.56 .
\end{gathered}
$$

From this, it follows that $m_{0}=23$.