Example 9. The germination rate of seeds of a certain crop is 0.75. Estimate the probability that out of 1000 sown seeds, the number of seeds that will germinate will be from 700 to 800 inclusive.
Solution. In this case, $M(X)=a=1000 \cdot 0.75=750$, the boundary values of the random variable $X$ are symmetric relative to $M(X)=750$. Therefore, from the original inequalities $700 \leq X \leq 800$, we can transition to the inequalities $-50 \leq X-750 \leq 50$, or $|X-a| \leq 50$, which gives the left part of inequality (4.1.4.) with $\varepsilon=50$. The value of $D(X)$ will be found using the formula $D(X)=n p q$, or $D(X)=1000 \cdot 0.75 \cdot 0.25=750 / 4$. Taking into account that $\varepsilon^{2}=50^{2}=2500$, we obtain the right part of inequality (4.1.4.):

$$
1-\frac{D(X)}{\varepsilon^{2}}=1-\frac{750}{4 \cdot 2500}=1-0.075=0.925
$$

Therefore,

$$
P(|X-750| \leq 50) \geq 0.925
$$