Example 11. What is the probability of event $A$ occurring in each trial, if the most probable number of occurrences of event $A$ in 120 trials is 32?
Solution. Given $n=120, k_{0}=32$. Inequalities (3.1.5) take the form

$$
120 p-(1-p) \leq 32 \leq 120 p+p
$$

or

$$
120 p-(1-p) \leq 32, \quad 120 p+p \geq 32
$$

Solving this system of inequalities, we find that

$$
\frac{32}{121} \leq p \leq \frac{33}{121}
$$