For most of the history of information retrieval (IR), search results were designed for human consumers who could scan, filter, and discard irrelevant information on their own. This shaped retrieval systems to optimize for finding and ranking more relevant documents, but not keeping results clean and minimal, as the human was the final filter. However, LLMs have changed that by lacking this filtering ability. To address this, we introduce Bits-over-Random (BoR), a chance-corrected measure of retrieval selectivity that reveals when high success rates mask random-level performance. We measure selectivity as $BoR = \log_{2}\left(\frac{\mathrm{P}_{obs}}{\mathrm{P}_{rand}}\right)$, where $\mathrm{P}_{rand}$ is the hypergeometric baseline for the chosen success rule (here, coverage: $ \geq1 $ relevant in top-$K$). On the 20 Newsgroups dataset, BM25 and SPLADE both report $>99$% success at $K=100$ (coverage), yet $BoR \approx 0$, indicating random-level selectivity at that depth. When the expected coverage ratio $\left(\frac{K \cdot \bar{R}_{q}}{N}\right)$ exceeds 3-5, the baseline dominates and selectivity collapses. Downstream retrieval-augmented generation (RAG) evaluation confirms this pattern: LLM accuracy can degrade substantially at $K=100$, consistent with the near-zero BoR ceiling. In contrast, BoR remains positive on BEIR/SciFact and on MS MARCO (where 41 systems cluster within 0.2 bits of the theoretical ceiling despite a 13-point recall gap), confirming baseline predictions across sparse and large-scale settings. We further show that the collapse boundary applies to LLM agent tool selection, where small catalog sizes cause selectivity to vanish even with perfect selectors. These findings suggest reporting BoR alongside traditional metrics and reconsidering depth choices when additional retrieval provides negligible selectivity gains while inflating computational costs.