We study the randomized version of a computation model (introduced by Grohe, Koch, and Schweikardt (ICALP'05); Grohe and Schweikardt (PODS'05)) that restricts random access to external memory and internal memory space. Essentially, this model can be viewed as a powerful version of a data stream model that puts no cost on sequential scans of external memory (as other models for data streams) and, in addition, (like other external memory models, but unlike streaming models), admits several large external memory devices that can be read and written to in parallel. We obtain tight lower bounds for the decision problems set equality, multiset equality, and checksort. More precisely, we show that any randomized one-sided-error bounded Monte Carlo algorithm for these problems must perform Omega(log N) random accesses to external memory devices, provided that the internal memory size is at most O(N^(1/4)/log N), where N denotes the size of the input data. From the lower bound on the set equality problem we can infer lower bounds on the worst case data complexity of query evaluation for the languages XQuery, XPath, and relational algebra on streaming data. More precisely, we show that there exist queries in XQuery, XPath, and relational algebra, such that any (randomized) Las Vegas algorithm that evaluates these queries must perform Omega(log N) random accesses to external memory devices, provided that the internal memory size is at most O(N^(1/4)/log N).