This involves a complex interplay of mathematical and statistical principles to determine such a threshold. Understanding this threshold is crucial for miners who are part of a mining pool and need to know when it might be beneficial to raise concerns about the distribution of tasks or rewards within the pool. The determination of this threshold would require an analysis of the mining pool's operations, including the distribution mechanism for tasks and rewards, the overall size of the pool, and the specific algorithms used to validate and include transactions in a block.

Moreover, the concept of Merkle inclusion is fundamental to how transactions are verified and included in a block on the blockchain. Each transaction generates a hash, and these hashes are paired, hashed together, and continually combined until a single hash remains, known as the Merkle root. This process ensures the integrity and verifiability of all transactions without needing to store the entire chain of transaction data. For a miner, challenging the pool over Merkle inclusion could imply discrepancies or disagreements over which transactions were included in a block or how they were selected and processed.

The decision for a miner to challenge the pool involves weighing the potential gains against the costs of such an action. This includes not only the immediate financial implications but also the possible long-term effects on their relationship with the pool and other miners. It is a strategic decision that must be supported by a clear understanding of the technical aspects of mining, as well as the policies and practices of the mining pool.

In summary, finding a precise mathematical or statistical threshold for when it is advantageous for a miner to challenge a pool over Merkle inclusion is a complex issue. It requires a deep understanding of blockchain technology, specifically how blocks and transactions are processed and validated. Additionally, it necessitates knowledge of the mining pool's operational protocols and the dynamics of participating in such a collective effort.