{"query": "Ten fair coins: the average and a particular outcome", "count": 20, "results": [{"id": "card_works_probability", "title": "Ten fair coins: the average and a particular outcome", "shelf": "the-works", "surface": "secular", "snippet": "Flip a fair coin ten times. On average five come up heads; the chance of exactly two heads is C(10,2)/2¹⁰. The engine confirms both the expectation and the exact probability.  Worked & sealed by the e", "authority_tier": "verified", "source": "The Works — worked & sealed", "generated": false}, {"id": "card_theory_measure_theory__lebesgue_integration", "title": "Measure theory (Lebesgue integration)", "shelf": "theories", "surface": "secular", "snippet": "Measure theory (Lebesgue integration) — an engine domain that can touch it: mathematics. Calibration: map-only — a foundation, not a computation. What it means for a set to have a SIZE, done carefully", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_src_word_coin_collection", "title": "coin collection", "shelf": "dictionary", "surface": "secular", "snippet": "coin collection: (noun) a collection of coins · (noun) the collection and study of money (and coins in particular) — syn: numismatics, numismatology, coin collecting", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_coin_collecting", "title": "coin collecting", "shelf": "dictionary", "surface": "secular", "snippet": "coin collecting: (noun) the collection and study of money (and coins in particular) — syn: numismatics, numismatology, coin collection", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_book_51302", "title": "English Coins and Tokens, with a Chapter on Greek and Roman Coins — Llewellynn Frederick William Jewitt", "shelf": "gutenberg", "surface": "secular", "snippet": "English Coins and Tokens, with a Chapter on Greek and Roman Coins, by Llewellynn Frederick William Jewitt. Subjects: Tokens -- Great Britain; Numismatics -- Great Britain; Coins, British. Read the ful", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_src_lex_g2772", "title": "G2772 — κέρμα (kerma): coin", "shelf": "lexicon", "surface": "secular", "snippet": "κέρμα (kerma), Strong's G2772: coin. κέρμα , -τος, τό (κείρω), __1. a slice, hence, __2. a small coin : pl., Jhn.2:15. † (AS)", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_alm_connection_entropy_is_thermodynamic", "title": "Almanac: Shannon entropy and thermodynamic entropy are the same quantity", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  The entropy of a message and the entropy of a gas are not analogies -- they are one formula. information_theory confirmed Shannon entropy numerically: a fair coin carries 1 bit, four equal", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_src_lex_g4715", "title": "G4715 — στατήρ (statēr): coin", "shelf": "lexicon", "surface": "secular", "snippet": "στατήρ (statēr), Strong's G4715: coin. στατήρ , -ῆρος, ὁ [in Aq., Sm., for מִשְׁקָל ;] a stater ; __(a) a weight ; __(b) a coin (used by late writers of the Greek τετράδραχμον): Mat.17:27 26:16 , WH, ", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_src_lex_g3546", "title": "G3546 — νόμισμα (nomisma): coin", "shelf": "lexicon", "surface": "secular", "snippet": "νόμισμα (nomisma), Strong's G3546: coin. νόμισμα , -τος, τό (νομίζω), [in LXX: Est.8:36 (דָּת), Neh.7:71 R (דַּרְכְּמוֹן), 1Ma.15:6 * ;] __1. that which is established by usage, a custom. __2. The cur", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_src_lex_g5475", "title": "G5475 — χαλκός (chalkos): copper/bronze/coin", "shelf": "lexicon", "surface": "secular", "snippet": "χαλκός (chalkos), Strong's G5475: copper/bronze/coin. χαλκός , -oῦ, ὁ [in LXX chiefly for נְחֹשֶׁת ;] copper : 1Co.13:1 , Rev.18:12 ; by meton., of copper coin, Mat.10:9 , Mrk.6:8 12:41 .† (AS)", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_src_lex_g3016", "title": "G3016 — λεπτός (leptos): coin", "shelf": "lexicon", "surface": "secular", "snippet": "λεπτός (leptos), Strong's G3016: coin. λεπτός , -ή, -όν (λέπω, to peel), __1. peeled. __2. fine, thin, small, light ; hence, in late Gk., as subst. τὸ λ., a small coin (one-eighth of an as, AV, mite):", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "generated": false}, {"id": "card_alm_connection_independence_multiplies_information_adds", "title": "Almanac: Independence multiplies probability but adds information", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  Flip two fair coins. The probability verifier confirmed that independent events MULTIPLY: P(A and B) = 0.5 x 0.5 = 0.25. The information_theory verifier confirmed that one fair coin carrie", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_src_book_75542", "title": "The coins of India — C. J. (Cecil Jermyn) Brown", "shelf": "gutenberg", "surface": "secular", "snippet": "The coins of India, by C. J. (Cecil Jermyn) Brown. Subjects: Numismatics -- India. Read the full text (public domain): https://www.gutenberg.org/ebooks/75542", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_src_book_61357", "title": "Coins de Paris — Georges Cain", "shelf": "gutenberg", "surface": "secular", "snippet": "Coins de Paris, by Georges Cain (in fr). Subjects: Paris (France) -- Description and travel. Read the full text (public domain): https://www.gutenberg.org/ebooks/61357", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_src_word_coin_bank", "title": "coin bank", "shelf": "dictionary", "surface": "secular", "snippet": "coin bank: (noun) a container (usually with a slot in the top) for keeping money at home — syn: savings bank, money box, bank", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_coin", "title": "coin", "shelf": "dictionary", "surface": "secular", "snippet": "coin: (noun) a flat metal piece (usually a disc) used as money · (verb) make up · (verb) form by stamping, punching, or printing — syn: mint, strike", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_book_33021", "title": "The Carlovingian Coins; Or, The Daughters of Charlemagne A Tale of the Ninth Century — Eugène Sue", "shelf": "gutenberg", "surface": "secular", "snippet": "The Carlovingian Coins; Or, The Daughters of Charlemagne A Tale of the Ninth Century, by Eugène Sue. Subjects: Charlemagne, Emperor, 742-814 -- Fiction. Read the full text (public domain): https://www", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_src_book_26063", "title": "A Coin of Edward VII: A Detective Story — Fergus Hume", "shelf": "gutenberg", "surface": "secular", "snippet": "A Coin of Edward VII: A Detective Story, by Fergus Hume. Subjects: Fiction; Detective and mystery stories. Read the full text (public domain): https://www.gutenberg.org/ebooks/26063", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}, {"id": "card_harmony_h066", "title": "The lost sheep, lost coin, and prodigal son (1 of 4 gospels)", "shelf": "scripture", "surface": "witness", "snippet": "The lost sheep, lost coin, and prodigal son — Later Judean and Perean Ministry. Recorded by 1 of the four gospels: Luke 15:1-32. Read them side by side at /harmony.html; the text is the World English ", "authority_tier": "reference", "source": "World English Bible (public domain) + the standard study tables", "generated": false}, {"id": "card_src_book_57020", "title": "The Regulating Silver Coin, Made Practicable and Easie, to the Government and Subject — Samuel Pratt", "shelf": "gutenberg", "surface": "secular", "snippet": "The Regulating Silver Coin, Made Practicable and Easie, to the Government and Subject, by Samuel Pratt. Subjects: Coinage -- Great Britain -- Early works to 1800; Great Britain. Parliament -- Early wo", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "generated": false}]}