A probabilistic approach to some of Euler's number theoretic identities

Probabilistic proofs and interpretations are given for the -binomial theorem, -binomial series, two of Euler's fundamental partition identities, and for -analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.