516. Independent trials are conducted with the same but unknown probability $p$ of event $A$ occurring in each trial. Find the confidence interval for estimating the probability $p$ with a reliability of 0.95, if in 60 trials event $A$ occurred 15 times.
Solution. According to the condition, $n=60, m=15, \gamma=0.95$. We find the relative frequency of the occurrence of event $A: w=m / n=15 / 60=0.25$.

We find $t$ from the relation $\Phi(t)=\gamma / 2=0.95 / 2=0.475$. From the Laplace function table (see Appendix 2), we find $t=1.96$.

We find the boundaries of the desired confidence interval:

$$
\begin{aligned}
& \rho_{1}=\frac{n}{t^{2}+n}\left[w+\frac{t^{2}}{2 n}-t \sqrt{\frac{w(1-w)}{n}+\left(\frac{t}{2 n}\right)^{2}}\right] \\
& \rho_{2}=\frac{n}{t^{2}+n}\left[w+\frac{t^{2}}{2 n}+t \sqrt{\frac{w(1-w)}{n}+\left(\frac{t}{2 n}\right)^{2}}\right]
\end{aligned}
$$

Substituting $n=60, w=0.25, t=1.96$ into these formulas, we get $p_{1}=0.16, p_{2}=0.37$.

Thus, the desired confidence interval is $0.16<p<0.37$.