Recursive Formula for the Random String Word Detection Probability, Overlaps and Probability Extremes
Bibliographic record
Abstract
In this paper, for the first time ever, the properties of the word detection probability in a random string have been investigated. The formerly known methods led to numerical evaluation of the researched probabilities only. The present work derives the simplest algorithm for calculation of the word’s at least once detection probability in a random string. A recursive formula that considers the overlap capability has been deduced for the probability under study. This formula is being used for the proposition on comparison of the word detection probabilities in a random string for the words with different periods. The result allows determining the structure of words that have maximum and minimum detection probabilities. In particular, words having equal number of alphabetic characters have been studied. It has been established, that for the words in question detection probability is minimal for the ideally symmetrical words that have irreducible period - and maximal for the words devoid of the overlap feature. These results will be useful for molecular genetics, as well as for students studying discrete mathematics, probability theory and molecular biology.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.020 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".