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Equation 27 · Part 5 · OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute

subtraction

pk=1−(1−p)k,p_k = 1 - (1-p)^k,
subtraction

What this part means

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

Its job in the formula

Subtract the following term or group from the preceding one. A leading minus marks a negative quantity.

The passage around this formula

The parallel-sampling case makes the shape of the returns explicit. If a single attempt succeeds with probability p and attempts were independent, the probability that at least one of k succeeds is pk=1−(1−p)kp_k = 1 - (1-p)^k. which is concave in k and saturates quickly. Two caveats destroy any naive extrapolation from it. Attempts from one model on one prompt are strongly correlated, so realised gains fall well below this bound; and pkp_k is only achievable if something can identify the successful attempt. Without a verifier, extra samples buy candidates, not answers. This is precisely why the reasoning-effort control and the availability of parallel test-time compute are architectural facts about a…

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Addition combines quantities; subtraction measures the signed difference between them. Parentheses show what is combined before the rest of the expression is evaluated.

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Sources cited in the surrounding passage

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