We will start with the coin analogy. It is in fact unclear how the analogy with coin tossing translates to cold-hit cases. Searching a larger database no doubt increases the probability of finding a match at some point, but is the increase as fast as the Arizona Department of Public Safety examples and the coin analogy suggest? Quite crucially, following (Donnelly & Friedman, 1999) we need to pay attention to what hypotheses are tested, what probabilistic methods the context recommends, and what exactly the evidence we obtained is. For instance, one hypothesis of
Donnelly, P., & Friedman, R. D. (1999). DNA Database Searches and the Legal Consumption of Scientific Evidence. Michigan Law Review, 97(4), 931. https://doi.org/10.2307/1290377