In evaluating an algorithm, worst-case analysis can be overly pessimistic. Average-case analysis can be overly optimistic. An intermediate approach is to show that an algorithm does well on a broad class of input distributions. Koutsoupias and Papadimitriou recently analyzed the least-recently-used (LRU) paging strategy in this manner, analyzing its performance on an input sequence generated by a so-called diffuse adversary -- one that must choose each request probabilitistically so that no page is chosen with probability more than some fixed epsilon>0. They showed that LRU achieves the optimal competitive ratio (for deterministic on-line algorithms), but they didn't determine the actual ratio. In this paper we estimate the optimal ratios within roughly a factor of two for both deterministic strategies (e.g. least-recently-used and first-in-first-out) and randomized strategies. Around the threshold epsilon ~ 1/k (where k is the cache size), the optimal ratios are both Theta(ln k). Below the threshold the ratios tend rapidly to O(1). Above the threshold the ratio is unchanged for randomized strategies but tends rapidly to Theta(k) for deterministic ones. We also give an alternate proof of the optimality of LRU.