Example 10. Starting from which number $n$ of independent trials is the inequality $p\left(\left|\frac{m}{n}-p\right|<0.1\right)>0.97$ satisfied, if in a single trial $p=0.8?$
Solution. Inequality (4.2.5) in this case takes the form

$$
P\left(\left|\frac{m}{n}-p\right|<0.01\right)>0.97
$$

From this we find

$$
1-0.97>\frac{0.16}{0.01 \cdot n}, 0.03>\frac{0.16}{0.01 \cdot n}, n>\frac{0.16}{0.03 \cdot 0.01}=\frac{0.16}{0.0003} \approx 533.33
$$

Thus, $n \geq 534$.

## Problems

1. The probability of a positive outcome in a single trial is $p=0.8$. Estimate the probability that in 1000 independent repeated trials, the deviation of the frequency of positive outcomes from the probability in a single trial in absolute value will be less than 0.05.
2. How many independent trials should be conducted so that the probability of the inequality $|m / n-p| \leq 0.06$ being satisfied exceeds 0.78, if the probability of the occurrence of the event in a single trial is $p=0.7$?
3. To determine the average burning duration of light bulbs in a batch of 200 identical boxes, one bulb was taken from each box as a sample. Estimate the probability that the average burning duration of the 200 selected light bulbs differs from the average burning duration of the entire batch in absolute value by less than 5 hours, given that the standard deviation of the burning duration of the bulbs in each box is less than 7 hours.
4. How many times should a given quantity, whose true value is $a$, be measured so that with a probability of at least 0.95, it can be asserted that the arithmetic mean of these measurements differs from $a$ in absolute value by less than 2, if the standard deviation of each measurement is less than 10?
5. How many independent measurements of a certain quantity should be made so that with a probability of at least 0.98, it can be asserted that the arithmetic mean of the measurement results differs from the true value in absolute value by less than 0.01, if the variance of a single measurement result does not exceed 1?
6. Given a sequence of independent random variables $X_{1}, X_{2}, \ldots, X_{n}, \ldots$ The random variable $X_{n}(n=1,2,3, \ldots)$ can take only three values: $-\sqrt{n}, 0, \sqrt{n}$ with probabilities equal to $\frac{1}{n}, 1-\frac{2}{n}, \frac{1}{n}$, respectively. Is Chebyshev's theorem applicable to this sequence of random variables?

## Answers

1. $P>0.936$. 2. $n>265$. 3. $P>0.99$. Hint. $D\left(X_{i}\right)<10$. If $n p q<10$, then these formulas lead to significant errors.

## Probability of Deviation of Relative Frequency from Constant Probability in Independent Trials.

The probability that in $n$ independent trials, in each of which the probability of the occurrence of an event is $p(0<p<1)$, the absolute deviation of the frequency of the occurrence of the event from the probability of the event does not exceed a positive number $\varepsilon$, is approximately equal to twice the value of the Laplace function at $x=\varepsilon \sqrt{n / p q}$:

$$
P\left(\left|\frac{m}{n}-p\right| \leq \varepsilon\right)=2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)
$$