Mention markets, a type of prediction market in which contracts resolve based on whether a specified keyword is mentioned during a future public event, require accurate probabilistic forecasts of keyword-mention outcomes. While recent work shows that large language models (LLMs) can generate forecasts competitive with human forecasters, it remains unclear how input context should be designed to support accurate prediction. In this paper, we study this question through experiments on earnings-call mention markets, which require forecasting whether a company will mention a specified keyword during its upcoming call. We run controlled comparisons varying (i) which contextual information is provided (news and/or prior earnings-call transcripts) and (ii) how \textit{market probability}, (i.e., prediction market contract price) is used. We introduce Market-Conditioned Prompting (MCP), which explicitly treats the market-implied probability as a prior and instructs the LLM to update this prior using textual evidence, rather than re-predicting the base rate from scratch. In our experiments, we find three insights: (1) richer context consistently improves forecasting performance; (2) market-conditioned prompting (MCP), which treats the market probability as a prior and updates it using textual evidence, yields better-calibrated forecasts; and (3) a mixture of the market probability and MCP (MixMCP) outperforms the market baseline. By dampening the LLM's posterior update with the market prior, MixMCP yields more robust predictions than either the market or the LLM alone.