I don't know exactly how to explain this, but if you have say N independent binary outcomes (say, 50 bits, where each bit is either on or off), and thus the probability distributions $p_{\text{i}}(x_{\text{i}}|y)$ are binomial, if you take the average probability $p_{x = 1}(y)$, you get something like the expected proportion of bits that are on for a given y. Considering all trials are independent, if we further assume a bit can be in only one state at any given instant time point t, if you multiply the average probability by the number of bits (50) and you round that product to the greatest integer lesser/equal to it, isn't that basically a (homogeneous) Poisson process?
Edit: In short, if you have N independent trials modelled by a binomial distribution, could the system taken as a whole be modelled as Poisson process?