Example 6.17. The probability of manufacturing a part of the highest grade on a given machine is 0.4. Find the probability that half of 26 randomly selected parts will be of the highest grade.
Solution. It is required to find the probability $P_{13,26}$. Knowing the Laplace theorem, the necessary calculations are performed according to the following scheme:

$$
\begin{aligned}
& p=0.4 ; \quad n p=26 \cdot 0.4=10.4 \\
& q=1-0.4=0.6 ; \quad n p q=10.4 \cdot 0.6=6.24 \\
& n=26 ; \quad \sqrt{n p q}=\sqrt{6.24}=2.50 ; \\
& m=13 ; \quad m-n p=13-10.4=2.6 \text {; } \\
& x=\frac{m-n p}{\sqrt{n p q}}=\frac{2.60}{2.50}=1.04 \\
& \phi(x)=\phi(1.04)=0.2323 \\
& P_{m n} \approx \frac{\phi(x)}{\sqrt{n p q}}=\frac{0.2323}{2.50}=0.093 .
\end{aligned}
$$

The values of the function $\phi(x)$ are found using a table.