Summary: We prove that the probability distribution of a random variable \(\xi\) represented in the form of an infinite series \[ \xi =\sum\limits_{k=1}^\infty\xi_k a_k \] is singular, where \(\xi_k\) are independent Bernoulli random variables and where the sequence \(\{ a_k\}\) is such that the series \(\sum^\infty_{k=1}a_k\) converges, \(a_k\geq 0\) for all \(k\geq 1\), and, for an arbitrary \(k\in\mathbb N\), there exists \(s_k\in\mathbb N\cup\{ 0\}\) for which \(s_k>0\) for infinitely many indices \(k\) and \(a_k=a_{k+1}=\cdots =a_{k+s_k}\geq r_{r_k+s_k}\), where \(r_k\) is the tail of the series, namely \[ r_k=\sum\limits_{i=k+1}^\infty a_i. \] Under these assumptions, it is shown that the corresponding distribution is a Bernoulli convolution with essential intersections (that is, almost all with respect to the Hausdorff-Besicovitch dimension points of the spectrum have continuum many different expansions of the form \(\sum^\infty_{k=1}\omega_ka_k\), where \(\omega_k\in\{ 0,1\}\)). Our main attention is paid to the studies of fractal properties of singularly continuous probability measures \(\mu_\xi\). In particular, fractal properties of the spectra (minimal closed supports of the above measures) and minimal in the sense of the Hausdorff-Besicovitch dimension dimensional supports of such probability distributions are studied in detail.