$p$-adic behavior of Bernoulli probabilities

We study the standard Bernoulli probabilistic scheme for independent random variables (the symmetric case). As usual, we are interested in limits of probabilities when the number of trails approaches infinity. However, these limits are considered with respect to the $p$-adic metric. This is a sufficiently exotic metric and it is surprising that ordinary (classical) probabilities have limits with respect to this metric. Thus we found a new asymptotic of the classical Bernoulli probabilities which was not visible before.