1713. 800 independent trials were conducted: in 200 of them, the probability of the expected outcome was 0.5; in 400 trials, this probability was 0.4; and, finally, in the remaining cases, the probability of a favorable outcome was 0.3. Estimate from below the probability that the deviation of the frequency of the expected outcome from the average probability in absolute value does not exceed 0.04.
Solution. First, we find the average probability of the event occurring in 800 trials. If $p_{i}$ denotes the probability of occurrence in the $i$-th trial, then

$$
\bar{p}=\frac{1}{n} \sum_{i=1}^{n} p_{i}=\frac{1}{800} \cdot(0.5 \cdot 200+0.4 \cdot 400+0.3 \cdot 200)=\frac{3.2}{8}=0.4
$$

The problem requires estimating from below the probability

$$
P\left(\left|\frac{m}{n}-0.4\right|<0.04\right)
$$

Using the Poisson theorem, we find the desired estimate

$$
P\left(\left|\frac{m}{n}-0.4\right|<0.04\right) \geqslant 1-\frac{0.5 \cdot(1-0.5) \cdot 200+0.4 \cdot 0.6 \cdot 400+0.3 \cdot 0.7 \cdot 200}{80^{2} \cdot(0.04)^{2}}=
$$

$$
=0.817 .
$$