Published equation contexts
Why this formula appears here
There is a simple reason a single successful transfer matters more than its raw success rate suggests. If one crafted probe defeats a given safety-trained policy with probability p , and successive attempts against it were independent, the probability that at least one of k attempts succeeds is . which climbs quickly even when p is small per attempt. Two caveats matter as much as the formula. Attempts against one fixed, deployed model are usually correlated rather than independent, so real gains from repeated probing fall below this bound; and a universal, transferable suffix is close to the case the formula flatters most, because it is a single artefact effective across…
Read the representative guide
Symbol k
k is part of the quantity the equation computes from the expression on the right.
Read this term in its guide →How to interpret it
Read it with the definitions, units, and assumptions supplied by the article.
Research cited beside this formula
Published contexts (3)
A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.
Equation 3 · Model Evaluation
Two Different Bets on How to Align a Frontier Model
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.
There is a simple reason a single successful transfer matters more than its raw success rate suggests. If one crafted probe defeats a given safety-trained policy with probability p , and successive attempts against it were independent, the probability that at least one of k attempts succeeds is . which climbs quickly even when p is small per attempt. Two caveats matter as much as the formula. Attempts against one fixed, deployed model are usually correlated rather than independent, so real gains from repeated probing fall below this bound; and a universal, transferable suffix is close to the case the formula flatters most, because it is a single artefact effective across…
Meanings in this article
Equation guide → · Article →Equation 3 · Model Evaluation
OpenAI and Claude on Formal Reasoning: What the Benchmarks Show, and Where They Mislead
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.
Consider what the o1 numbers alone expose about how these gains are produced. If a single attempt at a problem succeeds with probability p , and k attempts were independent, the probability that at least one succeeds is . a curve that climbs fast and then saturates. It is tempting to read o1’s jump from 74% (one sample) to 83% (64-sample consensus) to 93% (1,000-sample rerank) as roughly this shape. It is not, for two reasons that matter for how the number should be read. First, repeated attempts by one model on one problem are correlated — the same misconception that causes one failure tends to recur — so the realised gain from added samples falls well below what…
Meanings in this article
Equation guide → · Article →Equation 27 · Foundation Models
OpenAI Model Systems from First Principles: Weights, Post-Training, and Inference Compute
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.
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 . 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 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…