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Equation 27 · Part 3 · OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute

Symbol k

pk=1−(1−p)k,p_k = 1 - (1-p)^k,
kk

What this part means

k is part of the quantity the equation computes from the expression on the right.

Its job in the formula

k is part of the quantity the equation computes from the expression on the right.

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…

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Learn the underlying idea

A variable is a named place for a value. Its letter is a local label: x can mean position in one formula and a data point in another.

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Sources cited in the surrounding passage

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