243. The probability of event $A$ occurring in each trial is 1/2. Using Chebyshev's inequality,
estimate the probability that the number $X$ of occurrences of event $A$ is within the range from 40 to 60, if 100 independent trials are conducted.
Solution. Let's find the mathematical expectation and variance of the discrete random variable $X$ - the number of occurrences of event $A$ in 100 independent trials:

$$
M(X)=n p=100 \cdot 1 / 2=50 ; D(X)=n p q=100 \cdot 1 / 2 \cdot 1 / 2=25
$$

Let's find the maximum difference between the given number of occurrences of the event and the mathematical expectation $\boldsymbol{M}(\boldsymbol{X})=50$ :

$$
\varepsilon=60-50=10
$$

. We will use Chebyshev's inequality in the form

$$
P(|X-M(X)|<\varepsilon) \geqslant 1-D(X) / \varepsilon^{2}
$$

Substituting $M(X)=50, D(X)=25, \varepsilon=10$, we get

$$
P(|X-50|<10) \geqslant 1-25 / 10^{2}=0.75
$$