518. An experimental slot machine has been created, which is supposed to ensure a win in one out of 100 coin tosses into the machine. To test the suitability of the machine, 400 trials were conducted, during which the win appeared 5 times. Find the confidence interval that covers the unknown probability of a win with a reliability of $\gamma=0.999$.
Solution. Let's find the relative frequency of winning: $w=m / n=5 / 400=0.0125$. Let's find $t$ from the relation $\Phi(t)=$ $=\gamma / 2=0.999 / 2=0.4995$. From the table of the Lannace function (see Appendix 2), we find $t=3.3$.

Considering that $n=400$ is large, we will use approximate formulas to find the boundaries of the confidence interval:

$$
p_{1}=w-t \sqrt{w(1-w) / n}, \quad \rho_{2}=w+t \sqrt{w(1-w) / n} \text {. }
$$

Substituting $w=0.0125, t=3.3, n=400$ into these formulas, we get $p_{1}=-0.0058, p_{2}=0.0308$.

Thus, the required confidence interval is $0<p<0.0308$.