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If the condition $q(\hat{y}_{i+t}) > p(\hat{y}_{i+t})$ is met, the associated speculation is rejected with a probability calculated by the formula $1 - \frac{p(\hat{y}_{i+t})}{q(\hat{y}_{i+t})}$.

Probability Formula for Speculation Rejection

A system uses a fast, approximate model (generating probability distribution 'q') to propose outputs, which are then checked against a slower, more accurate model (generating probability distribution 'p'). An output is only considered for rejection if its probability from the fast model is greater than its probability from the slow model. Given the data below, determine which proposed output ('Token X' or 'Token Y') is a candidate for this rejection process, and calculate its specific rejection probability. Explain your reasoning for both tokens.

Speculative Rejection Scenario

Consider a scenario where a potential future value is being evaluated. The probability of this value under distribution 'p' is always less than its probability under distribution 'q'. Explain what happens to the probability of rejecting this value as the probability from 'p' gets closer to the probability from 'q'. Justify your answer by referencing the components of the rejection formula.

Analyzing the Speculation Rejection Formula

A predictive system is evaluating a potential output. The probability of this output under distribution 'p' is 0.2, and the probability under distribution 'q' is 0.8. Given that the probability from 'q' is greater than the probability from 'p', what is the probability that this potential output will be rejected?

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