{"id": "card_works_probability", "kind": "verified", "title": "Ten fair coins: the average and a particular outcome", "body": "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 engine — E[Binom(10, ½)] = n·p = 5 · P(X=2) = C(10,2)·½¹⁰ = 45/1024 ≈ 0.04395.  [HOLDS; open the seal to re-check.]", "source": {"label": "The Works — worked & sealed", "url": "https://narrowhighway.com/s/e50261aa12a5d659f1567480f4802f5156bdb9e44fbb190a17e3f0d8e557d34f", "domain": "probability", "authority_tier": "verified"}, "shelf": "the-works", "box": "mathematics", "bands": ["the-works", "mathematics", "probability", "probability"], "connections": [{"to_card_id": "card_spine_the-works", "relationship": "member_of", "evidence": "a member of the the-works shelf in the keeping"}, {"to_card_id": "card_k_floor_of_discovery", "relationship": "paves", "evidence": "A worked, engine-sealed demonstration — a paving-stone of the floor of reality."}, {"to_card_id": "card_v_e25edb860b45", "relationship": "demonstrates", "evidence": "A worked demonstration in the same field (probability)."}, {"to_card_id": "card_v_5b4a7cb1d159", "relationship": "demonstrates", "evidence": "A worked demonstration in the same field (probability)."}, {"to_card_id": "card_works_bayes", "relationship": "kindred", "evidence": "A sibling worked demonstration in probability."}, {"to_card_id": "card_works_linear_algebra", "relationship": "adjacent", "evidence": "Neighbouring worked demonstrations in the same part of the volume."}, {"to_card_id": "card_works_information", "relationship": "adjacent", "evidence": "Neighbouring worked demonstrations in the same part of the volume."}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "subject": "Probability", "extra": {"seal_hash": "e50261aa12a5d659f1567480f4802f5156bdb9e44fbb190a17e3f0d8e557d34f", "cite_url": "https://narrowhighway.com/s/e50261aa12a5d659f1567480f4802f5156bdb9e44fbb190a17e3f0d8e557d34f", "demonstration_id": "works_probability", "verdict": "HOLDS"}, "presentation": {"glyph": "•", "kind_label": "mathematics", "by": "The Works — worked & sealed", "authority": "verified", "posted": "", "standing": "", "link": {"url": "https://narrowhighway.com/s/e50261aa12a5d659f1567480f4802f5156bdb9e44fbb190a17e3f0d8e557d34f", "host": "narrowhighway.com", "provider": "", "titled": "Ten fair coins: the average and a particular outcome", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_the-works", "title": "The Works — sealed, worked demonstrations", "relationship": "on the shelf of", "why": "a member of the the-works shelf in the keeping", "href": "/card/card_spine_the-works", "resolved": true}, {"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "paves", "why": "A worked, engine-sealed demonstration — a paving-stone of the floor of reality.", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_v_e25edb860b45", "title": "E[Binom(10, ½)] = n·p = 5 · P(X=2) = C(10,2)·½¹⁰ = 45/1024 ≈ 0.04395", "relationship": "demonstrates", "why": "A worked demonstration in the same field (probability).", "href": "/card/card_v_e25edb860b45", "resolved": true}, {"id": "card_v_5b4a7cb1d159", "title": "P(ill | +) = (.9·.01)/(.9·.01 + .05·.99) ≈ 0.154", "relationship": "demonstrates", "why": "A worked demonstration in the same field (probability).", "href": "/card/card_v_5b4a7cb1d159", "resolved": true}, {"id": "card_works_bayes", "title": "The base rate a doctor must not forget", "relationship": "kindred", "why": "A sibling worked demonstration in probability.", "href": "/card/card_works_bayes", "resolved": true}, {"id": "card_works_linear_algebra", "title": "Vectors and matrices: a dot product and a determinant", "relationship": "adjacent", "why": "Neighbouring worked demonstrations in the same part of the volume.", "href": "/card/card_works_linear_algebra", "resolved": true}, {"id": "card_works_information", "title": "Measuring information: one bit, and a distance", "relationship": "adjacent", "why": "Neighbouring worked demonstrations in the same part of the volume.", "href": "/card/card_works_information", "resolved": true}]}