When Does a New Purchase (or a New Word) Appear? Exact Probability Theory, Validated by Simulation, and Tested Against Real Text Data

Given m categories drawn independently and identically with probabilities P1,...,Pm, the probability that the n-th draw is of a category not seen in the previous n-1 draws is Pr(n-th new) = Σ_i P_i(1-P_i)^(n-1). We first verify this formula and its close relative, the expected number of distinct categories seen after n draws, by exact computation and independent Monte Carlo simulation. We then illustrate the formula's underlying i.i.d. assumption against real data: treating each distinct word in three Shakespeare plays (67,908 tokens, 7,797 distinct types) as a "category" with probability equal to its empirical frequency, we compare the true vocabulary-growth curve against a matched i.i.d. multinomial simulation. The real sequence deviates from the i.i.d. prediction by a large and highly statistically significant margin at every point tested: vocabulary grows 9-16% slower than the i.i.d. formula predicts through roughly the first two-thirds of each play, then overtakes in the final third. We emphasize that the underlying phenomena — word burstiness and sublinear vocabulary growth — are extensively studied and already explained by sophisticated generative models. This note's contribution is narrower: an independently verified, closed-form illustration of the same qualitative phenomenon, obtained by connecting it to the classical new-type-arrival formula, together with a specific, quantified two-regime decomposition of the deviation.