{
  "created": "2026-09-22",
  "python": "3.12.14",
  "numpy": "2.3.5",
  "generator": "NumPy PCG64",
  "base_seed": 20260922,
  "license": "CC BY 4.0 for aggregate datasets; MIT for code",
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      "title": "Ten million 6/49 draws: how uneven does fair look?",
      "seed": 20260923,
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      "limits": "A simulated fair process cannot certify a real lottery. Individual counts are binomial; counts for different numbers are dependent. The maximum is selected after inspecting 49 counts.",
      "csv": "ten-million-649-draws.csv",
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        [
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        [
          1883,
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          0.7
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        [
          1884,
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        [
          1885,
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        [
          1886,
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          0.75,
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        [
          1887,
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        [
          1888,
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          0.45,
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        [
          1889,
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        [
          1890,
          1.0,
          0.95,
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        [
          1891,
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          0.7,
          0.8
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        [
          1892,
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        [
          1893,
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          0.75
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        [
          1894,
          0.5,
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          0.8
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        [
          1895,
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        [
          1896,
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          0.6,
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        [
          1897,
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        [
          1898,
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          0.65,
          0.6
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        [
          1899,
          0.85,
          0.95,
          0.9
        ],
        [
          1900,
          0.85,
          0.65,
          0.75
        ],
        [
          1901,
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          0.4,
          0.6
        ],
        [
          1902,
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          0.5,
          0.75
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        [
          1903,
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        [
          1904,
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          0.8
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        [
          1905,
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          1.0,
          0.7
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        [
          1906,
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        [
          1907,
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          0.7,
          0.95
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        [
          1908,
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        [
          1909,
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        [
          1910,
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          0.4,
          0.7
        ],
        [
          1911,
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          0.65
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        [
          1912,
          0.8,
          0.6,
          0.6
        ],
        [
          1913,
          1.05,
          0.75,
          0.75
        ],
        [
          1914,
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          0.55,
          0.65
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        [
          1915,
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          0.9
        ],
        [
          1916,
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          1.15,
          0.75
        ],
        [
          1917,
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          0.8,
          0.8
        ],
        [
          1918,
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          0.9,
          0.8
        ],
        [
          1919,
          0.9,
          1.05,
          0.55
        ],
        [
          1920,
          0.55,
          0.95,
          0.7
        ],
        [
          1921,
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          0.55,
          0.8
        ],
        [
          1922,
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          0.7,
          0.75
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        [
          1923,
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          0.9,
          0.9
        ],
        [
          1924,
          0.65,
          0.95,
          0.85
        ],
        [
          1925,
          0.55,
          0.5,
          1.05
        ],
        [
          1926,
          0.8,
          0.75,
          0.65
        ],
        [
          1927,
          0.95,
          0.9,
          0.7
        ],
        [
          1928,
          0.6,
          0.55,
          0.6
        ],
        [
          1929,
          0.7,
          0.6,
          0.65
        ],
        [
          1930,
          0.85,
          0.85,
          0.85
        ],
        [
          1931,
          0.7,
          0.8,
          0.9
        ],
        [
          1932,
          0.85,
          0.9,
          0.7
        ],
        [
          1933,
          0.65,
          0.6,
          1.05
        ],
        [
          1934,
          0.85,
          0.65,
          0.85
        ],
        [
          1935,
          0.75,
          0.6,
          0.6
        ],
        [
          1936,
          0.65,
          0.75,
          0.9
        ],
        [
          1937,
          0.8,
          0.65,
          0.5
        ],
        [
          1938,
          0.35,
          0.7,
          1.05
        ],
        [
          1939,
          1.05,
          1.0,
          0.35
        ],
        [
          1940,
          0.75,
          0.7,
          1.15
        ],
        [
          1941,
          0.85,
          0.65,
          0.75
        ],
        [
          1942,
          0.3,
          1.1,
          0.8
        ],
        [
          1943,
          0.8,
          0.45,
          0.95
        ],
        [
          1944,
          0.85,
          0.65,
          0.55
        ],
        [
          1945,
          0.5,
          0.9,
          0.75
        ],
        [
          1946,
          0.8,
          0.85,
          0.8
        ],
        [
          1947,
          0.75,
          0.75,
          0.65
        ],
        [
          1948,
          0.95,
          0.5,
          0.45
        ],
        [
          1949,
          0.7,
          0.8,
          0.35
        ],
        [
          1950,
          1.1,
          0.7,
          0.8
        ],
        [
          1951,
          0.9,
          0.5,
          0.55
        ],
        [
          1952,
          0.6,
          0.8,
          0.6
        ],
        [
          1953,
          0.5,
          0.9,
          0.55
        ],
        [
          1954,
          0.7,
          0.45,
          0.65
        ],
        [
          1955,
          0.6,
          0.6,
          0.8
        ],
        [
          1956,
          0.9,
          0.65,
          0.95
        ],
        [
          1957,
          0.95,
          0.65,
          0.65
        ],
        [
          1958,
          1.1,
          0.55,
          0.95
        ],
        [
          1959,
          0.95,
          0.8,
          0.9
        ],
        [
          1960,
          0.6,
          0.8,
          0.85
        ],
        [
          1961,
          0.65,
          0.95,
          0.6
        ],
        [
          1962,
          0.8,
          0.65,
          0.95
        ],
        [
          1963,
          0.75,
          0.8,
          0.55
        ],
        [
          1964,
          0.65,
          1.0,
          0.75
        ],
        [
          1965,
          0.95,
          0.6,
          0.8
        ],
        [
          1966,
          0.6,
          0.9,
          0.5
        ],
        [
          1967,
          0.4,
          0.4,
          0.6
        ],
        [
          1968,
          0.6,
          1.05,
          0.65
        ],
        [
          1969,
          0.45,
          1.0,
          1.05
        ],
        [
          1970,
          0.8,
          1.15,
          0.85
        ],
        [
          1971,
          0.75,
          0.9,
          0.8
        ],
        [
          1972,
          0.6,
          0.5,
          0.95
        ],
        [
          1973,
          0.85,
          0.75,
          0.85
        ],
        [
          1974,
          0.4,
          0.6,
          0.85
        ],
        [
          1975,
          1.1,
          0.35,
          0.7
        ],
        [
          1976,
          1.0,
          0.75,
          0.75
        ],
        [
          1977,
          0.65,
          1.0,
          0.75
        ],
        [
          1978,
          0.65,
          0.55,
          0.8
        ],
        [
          1979,
          0.65,
          1.35,
          0.9
        ],
        [
          1980,
          0.75,
          0.75,
          0.75
        ],
        [
          1981,
          0.45,
          1.15,
          0.45
        ],
        [
          1982,
          0.4,
          0.85,
          0.7
        ],
        [
          1983,
          0.8,
          0.55,
          0.85
        ],
        [
          1984,
          0.5,
          0.9,
          0.65
        ],
        [
          1985,
          1.05,
          0.85,
          0.8
        ],
        [
          1986,
          0.85,
          0.65,
          0.55
        ],
        [
          1987,
          0.75,
          0.55,
          0.75
        ],
        [
          1988,
          1.0,
          0.6,
          0.65
        ],
        [
          1989,
          0.75,
          0.65,
          0.55
        ],
        [
          1990,
          0.45,
          0.9,
          0.7
        ],
        [
          1991,
          0.7,
          0.7,
          0.85
        ],
        [
          1992,
          0.6,
          0.75,
          0.6
        ],
        [
          1993,
          0.65,
          0.7,
          0.8
        ],
        [
          1994,
          0.65,
          0.6,
          0.45
        ],
        [
          1995,
          0.85,
          0.7,
          0.5
        ],
        [
          1996,
          0.55,
          1.35,
          1.05
        ],
        [
          1997,
          0.75,
          0.6,
          0.8
        ],
        [
          1998,
          0.7,
          0.7,
          0.85
        ],
        [
          1999,
          0.65,
          0.95,
          0.8
        ],
        [
          2000,
          0.65,
          0.7,
          0.65
        ],
        [
          2001,
          0.7,
          0.8,
          0.75
        ],
        [
          2002,
          0.65,
          0.85,
          0.65
        ],
        [
          2003,
          0.55,
          0.55,
          0.55
        ],
        [
          2004,
          0.6,
          0.65,
          0.75
        ],
        [
          2005,
          0.9,
          0.5,
          0.75
        ],
        [
          2006,
          0.95,
          0.5,
          0.9
        ],
        [
          2007,
          1.0,
          0.8,
          0.65
        ],
        [
          2008,
          0.75,
          0.7,
          0.95
        ],
        [
          2009,
          0.7,
          0.65,
          0.7
        ],
        [
          2010,
          0.85,
          0.45,
          0.45
        ],
        [
          2011,
          0.75,
          0.9,
          0.85
        ],
        [
          2012,
          0.5,
          0.85,
          0.75
        ],
        [
          2013,
          0.6,
          0.75,
          0.7
        ],
        [
          2014,
          0.8,
          0.85,
          0.7
        ],
        [
          2015,
          0.65,
          0.65,
          0.4
        ],
        [
          2016,
          0.75,
          1.0,
          0.5
        ],
        [
          2017,
          0.85,
          0.55,
          0.95
        ],
        [
          2018,
          0.8,
          0.55,
          0.9
        ],
        [
          2019,
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          0.95,
          0.95
        ],
        [
          2020,
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          0.55,
          0.6
        ],
        [
          2021,
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          0.7,
          0.8
        ],
        [
          2022,
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          0.6
        ],
        [
          2023,
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          0.75,
          0.75
        ],
        [
          2024,
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          0.8,
          0.6
        ],
        [
          2025,
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          0.55,
          0.75
        ],
        [
          2026,
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          0.6,
          0.45
        ],
        [
          2027,
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          0.55,
          0.75
        ],
        [
          2028,
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          0.75,
          1.15
        ],
        [
          2029,
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          0.7,
          0.7
        ],
        [
          2030,
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          0.4
        ],
        [
          2031,
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          0.75,
          0.65
        ],
        [
          2032,
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          0.4,
          0.75
        ],
        [
          2033,
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          0.7,
          1.0
        ],
        [
          2034,
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          0.9,
          0.55
        ],
        [
          2035,
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          1.0,
          0.85
        ],
        [
          2036,
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          0.95
        ],
        [
          2037,
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          0.9,
          0.7
        ],
        [
          2038,
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          0.8,
          0.65
        ],
        [
          2039,
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          0.4,
          0.7
        ],
        [
          2040,
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          1.0,
          0.7
        ],
        [
          2041,
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          0.5
        ],
        [
          2042,
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          0.65
        ],
        [
          2043,
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        ],
        [
          2044,
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        ],
        [
          2045,
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        ],
        [
          2046,
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          0.65
        ],
        [
          2047,
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        ],
        [
          2048,
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        ],
        [
          2049,
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        ],
        [
          2050,
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        ],
        [
          2051,
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        ],
        [
          2052,
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        ],
        [
          2053,
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        ],
        [
          2054,
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        ],
        [
          2055,
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        [
          2056,
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        ],
        [
          2057,
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        [
          2058,
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        ],
        [
          2059,
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        [
          2060,
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        [
          2061,
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        ],
        [
          2062,
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        [
          2063,
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        [
          2064,
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        [
          2065,
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        [
          2066,
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        [
          2067,
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        [
          2068,
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        ],
        [
          2069,
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        [
          2070,
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        ],
        [
          2071,
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        [
          2072,
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        ],
        [
          2073,
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        ],
        [
          2074,
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        [
          2075,
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        [
          2076,
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        ],
        [
          2077,
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        ],
        [
          2078,
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        [
          2079,
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        ],
        [
          2080,
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        ],
        [
          2081,
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        ],
        [
          2082,
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        ],
        [
          2083,
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        ],
        [
          2084,
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        ],
        [
          2085,
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        ],
        [
          2086,
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        ],
        [
          2087,
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        ],
        [
          2088,
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        ],
        [
          2089,
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        ],
        [
          2090,
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          0.8
        ],
        [
          2091,
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        ],
        [
          2092,
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        ],
        [
          2093,
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          0.7
        ],
        [
          2094,
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          0.55,
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        ],
        [
          2095,
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        ],
        [
          2096,
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          0.9
        ],
        [
          2097,
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          0.7,
          1.15
        ],
        [
          2098,
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          1.05
        ],
        [
          2099,
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          0.65,
          0.85
        ],
        [
          2100,
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          0.45,
          0.95
        ],
        [
          2101,
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          0.75,
          0.75
        ],
        [
          2102,
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          1.1,
          0.65
        ],
        [
          2103,
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          0.55,
          0.75
        ],
        [
          2104,
          0.85,
          0.55,
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        ],
        [
          2105,
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          0.95,
          0.75
        ],
        [
          2106,
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          0.85
        ],
        [
          2107,
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        ],
        [
          2108,
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        ],
        [
          2109,
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        ],
        [
          2110,
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        ],
        [
          2111,
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          0.7
        ],
        [
          2112,
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          0.8
        ],
        [
          2113,
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          0.9
        ],
        [
          2114,
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        ],
        [
          2115,
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        ],
        [
          2116,
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        ],
        [
          2117,
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        ],
        [
          2118,
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          0.75
        ],
        [
          2119,
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          0.95
        ],
        [
          2120,
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          0.6,
          0.55
        ],
        [
          2121,
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        ],
        [
          2122,
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          0.75
        ],
        [
          2123,
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        ],
        [
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          0.65,
          0.8
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          0.7,
          0.95
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          0.55,
          0.85,
          0.45
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          0.65,
          0.75,
          0.75
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        [
          19925,
          0.7,
          0.9,
          0.9
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        [
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          0.6,
          0.65
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        [
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          0.85,
          0.7
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        [
          19928,
          1.15,
          0.75,
          0.55
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        [
          19929,
          0.4,
          1.0,
          0.55
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        [
          19930,
          0.6,
          0.9,
          0.5
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        [
          19931,
          0.5,
          0.95,
          0.5
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        [
          19932,
          0.9,
          0.9,
          0.55
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        [
          19933,
          1.1,
          0.5,
          1.0
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        [
          19934,
          0.75,
          0.5,
          1.35
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        [
          19935,
          0.45,
          0.55,
          0.6
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        [
          19936,
          0.6,
          1.2,
          1.05
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        [
          19937,
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          0.8
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        [
          19938,
          0.55,
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          0.6
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        [
          19939,
          0.8,
          1.05,
          0.85
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        [
          19940,
          1.05,
          0.7,
          1.3
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        [
          19941,
          0.55,
          0.75,
          0.9
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        [
          19942,
          0.65,
          0.75,
          1.15
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        [
          19943,
          0.8,
          0.4,
          0.6
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        [
          19944,
          0.95,
          0.65,
          1.0
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        [
          19945,
          0.45,
          0.95,
          0.65
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        [
          19946,
          0.65,
          0.95,
          1.15
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        [
          19947,
          0.6,
          0.95,
          0.9
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        [
          19948,
          0.95,
          0.55,
          0.9
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        [
          19949,
          0.8,
          0.95,
          0.7
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        [
          19950,
          0.75,
          0.7,
          0.8
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        [
          19951,
          0.85,
          0.55,
          0.9
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        [
          19952,
          0.6,
          0.85,
          0.7
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        [
          19953,
          0.5,
          0.8,
          0.55
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        [
          19954,
          0.6,
          0.5,
          0.9
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        [
          19955,
          0.6,
          0.7,
          0.6
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        [
          19956,
          0.95,
          0.75,
          0.6
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        [
          19957,
          0.65,
          0.8,
          0.9
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        [
          19958,
          0.6,
          0.5,
          0.65
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        [
          19959,
          0.6,
          1.0,
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        [
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          0.75
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        [
          19961,
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          0.8
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        [
          19962,
          0.95,
          0.55,
          0.7
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        [
          19963,
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          0.7
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        [
          19964,
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          0.65,
          0.5
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        [
          19965,
          0.85,
          0.7,
          0.95
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        [
          19966,
          0.55,
          0.75,
          0.8
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        [
          19967,
          0.65,
          0.7,
          0.65
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        [
          19968,
          0.8,
          0.6,
          0.55
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        [
          19969,
          0.75,
          0.9,
          0.9
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        [
          19970,
          0.9,
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          0.9
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        [
          19971,
          0.85,
          0.8,
          0.55
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        [
          19972,
          0.85,
          0.6,
          0.55
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        [
          19973,
          0.85,
          0.6,
          0.85
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        [
          19974,
          1.0,
          0.9,
          0.75
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        [
          19975,
          0.55,
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          0.65
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        [
          19976,
          0.65,
          0.75,
          0.85
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        [
          19977,
          0.65,
          0.85,
          0.7
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        [
          19978,
          0.9,
          0.4,
          0.85
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        [
          19979,
          0.9,
          0.6,
          0.7
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        [
          19980,
          0.6,
          0.8,
          0.7
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        [
          19981,
          0.5,
          1.15,
          0.75
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        [
          19982,
          0.6,
          0.9,
          0.9
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        [
          19983,
          0.85,
          0.45,
          0.9
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        [
          19984,
          0.6,
          0.6,
          0.8
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        [
          19985,
          0.75,
          0.75,
          0.55
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        [
          19986,
          0.4,
          0.75,
          0.7
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        [
          19987,
          0.75,
          1.05,
          0.55
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        [
          19988,
          0.9,
          0.8,
          0.4
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        [
          19989,
          0.65,
          0.95,
          0.7
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        [
          19990,
          0.65,
          0.65,
          0.8
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        [
          19991,
          0.6,
          0.65,
          0.8
        ],
        [
          19992,
          0.6,
          1.1,
          0.8
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        [
          19993,
          1.05,
          1.0,
          0.8
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        [
          19994,
          0.8,
          0.95,
          0.9
        ],
        [
          19995,
          0.6,
          0.8,
          0.4
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        [
          19996,
          0.55,
          0.65,
          0.9
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        [
          19997,
          0.75,
          0.95,
          0.8
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        [
          19998,
          0.7,
          0.95,
          0.9
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        [
          19999,
          0.7,
          0.85,
          0.45
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        [
          20000,
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          0.9,
          0.8
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      ],
      "sha256": "ed03a4edafb2a02bf832b392a8410373db15011502ab6933256d803722d77e70"
    },
    {
      "id": 3,
      "slug": "ordered-versus-irregular",
      "title": "1-2-3 versus 2-6-9: exact enumeration and one million draws",
      "seed": 20260925,
      "summary": {
        "draws": 1000000,
        "combinations": 120,
        "ordered": 8209,
        "irregular": 8294
      },
      "method": "Enumerate C(10,3)=120 subsets exactly, then sample 1,000,000 uniform 3/10 draws. Compare two combinations chosen before simulation. This is explicitly a reduced model, not a 6/49 jackpot experiment.",
      "limits": "Equal probability is established by counting. Finite sample counts will differ. A million 6/49 trials usually gives zero observations of a specified combination, so it would be a poor demonstration.",
      "csv": "ordered-versus-irregular.csv",
      "columns": [
        "combination",
        "observed",
        "expected"
      ],
      "rows": [
        [
          "1-2-3",
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          "6-7-10",
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          "6-8-10",
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          "6-9-10",
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        [
          "7-8-9",
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        [
          "7-8-10",
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          "7-9-10",
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          "8-9-10",
          8364,
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      ],
      "sha256": "a955135f7c19ab61819622e749a01c4ef11027c1256780f08d59b98de8682d4e"
    },
    {
      "id": 4,
      "slug": "overdue-gaps",
      "title": "How long can one number stay absent? A million complete waits",
      "seed": 20260926,
      "summary": {
        "waits": 1000000,
        "mean_misses": 7.15402,
        "theory_mean_misses": 7.166666666666667,
        "longest_observed": 110
      },
      "method": "Draw one million independent geometric waiting times for a prespecified number with inclusion probability 6/49. Subtract one so each observation counts missed draws before success. All waits are complete, avoiding right censoring.",
      "limits": "The longest wait among a million is not a forecast for the next wait. This model describes one fixed number, not the maximum gap over 49 dependent number histories.",
      "csv": "overdue-gaps.csv",
      "columns": [
        "minimum_misses",
        "maximum_misses",
        "observed",
        "expected"
      ],
      "rows": [
        [
          0,
          4,
          480027,
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        [
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          9,
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        [
          10,
          19,
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        [
          20,
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        [
          40,
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        [
          80,
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      ],
      "sha256": "dd59432b7bd70c925fbde92195ffc58309a1994ebe8fccb4bb019879df59ff4b"
    },
    {
      "id": 5,
      "slug": "consecutive-pairs",
      "title": "Consecutive numbers: the exact answer meets a million draws",
      "seed": 20260927,
      "summary": {
        "draws": 1000000,
        "at_least_one_pair": 0.494678,
        "exact_probability": 0.4951984494075151
      },
      "method": "Count adjacent differences equal to one in each sorted 6/49 draw. A run of three contributes two adjacent pairs. Compare the fraction with at least one pair to 1 \u2212 C(44,6)/C(49,6).",
      "limits": "Pairs within a draw are not independent. This is not a count of separate maximal consecutive runs, and a likely category does not improve any individual combination.",
      "csv": "consecutive-pairs.csv",
      "columns": [
        "adjacent_pairs",
        "observed"
      ],
      "rows": [
        [
          0,
          505322
        ],
        [
          1,
          388311
        ],
        [
          2,
          96777
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        [
          3,
          9248
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        [
          4,
          340
        ],
        [
          5,
          2
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      ],
      "sha256": "00868f20712696a622e4c30ce9359217e8824936a2befbca91dcfca06a134770"
    },
    {
      "id": 6,
      "slug": "overlap-between-draws",
      "title": "Two fresh draws, one million times: how many numbers repeat?",
      "seed": 20260928,
      "summary": {
        "pairs": 1000000,
        "mean_overlap": 0.733503,
        "theory_mean": 0.7346938775510204
      },
      "method": "Generate one million independent pairs of 6/49 draws. Count the size of their intersection. Pairing disjoint draws makes the experimental trials independent.",
      "limits": "This measures shared numbers, not identical ordered extraction sequences. The full-repeat category is too rare for stable relative precision at this sample size.",
      "csv": "overlap-between-draws.csv",
      "columns": [
        "matches",
        "observed",
        "expected"
      ],
      "rows": [
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    {
      "id": 7,
      "slug": "parity-categories",
      "title": "Balanced groups, equal tickets: the odd\u2013even experiment",
      "seed": 20260929,
      "summary": {
        "draws": 1000000,
        "three_odd_fraction": 0.331666,
        "all_odd_fraction": 0.012827
      },
      "method": "Group one million uniform 6/49 draws by odd count. Compare each category with C(25,r)C(24,6\u2212r)/C(49,6).",
      "limits": "A category contains many combinations. Comparing categories does not compare the probabilities of two specified tickets.",
      "csv": "parity-categories.csv",
      "columns": [
        "odd_count",
        "observed",
        "expected"
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      "rows": [
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    {
      "id": 8,
      "slug": "birthday-collisions",
      "title": "Why any repeat arrives sooner than one chosen repeat",
      "seed": 20260930,
      "summary": {
        "trials": 10000,
        "outcomes": 10000,
        "median_first_repeat": 118.0
      },
      "method": "Run 10,000 independent sequences on 10,000 equally likely outcomes, stopping at the first repeated outcome. Calculate exact collision probabilities with the product of (1\u2212j/M).",
      "limits": "The reduced state space makes the birthday effect visible. It is not a claim that a particular real lottery combination repeats after 118 draws.",
      "csv": "birthday-collisions.csv",
      "columns": [
        "draws",
        "observed_collision_probability",
        "exact_probability"
      ],
      "rows": [
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          25,
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        [
          50,
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          100,
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          118,
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          150,
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          200,
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      "sha256": "c3b955f884cf7650b7b5f94d76806ddaed6a9378d1928d0bf1619fbf2786ec17"
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    {
      "id": 9,
      "slug": "multiple-testing",
      "title": "Twenty chances to be fooled: a false-positive experiment",
      "seed": 20260931,
      "summary": {
        "experiments": 100000,
        "tests_per_experiment": 20,
        "any_false_positive": 0.64015,
        "exact": 0.6415140775914581
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      "method": "Model 20 independent tests with exactly calibrated 5% false-positive rates as Bernoulli indicators, repeated 100,000 times. This isolates the multiple-comparisons mechanism.",
      "limits": "Actual lottery tests often share data and are dependent; 1\u22120.95^20 is then not exact. These are simulated significance indicators, not computed p-values from historical draw data.",
      "csv": "multiple-testing.csv",
      "columns": [
        "false_positive_count",
        "experiments"
      ],
      "rows": [
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        [
          4,
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        [
          5,
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        [
          6,
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        [
          7,
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        [
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        [
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        [
          10,
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        [
          11,
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        [
          12,
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        [
          13,
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        [
          14,
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          15,
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          16,
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          17,
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          18,
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          19,
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          20,
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      "sha256": "dcde88eeabb849c4f3b29366a0f4ad5d12e267d941cea5ba11c57f90b8d3c59c"
    },
    {
      "id": 10,
      "slug": "law-large-numbers",
      "title": "Closer in proportion, further in count: the law of large numbers",
      "seed": 20260932,
      "summary": {
        "replicates_per_size": 20000
      },
      "method": "Generate 20,000 independent binomial counts at each sample size. Report mean absolute deviations and theoretical standard deviations separately; they are different summaries.",
      "limits": "Different sample sizes use independent replicates, not one continuing trajectory. Average relative error shrinks; no individual path is forced to improve at every step.",
      "csv": "law-large-numbers.csv",
      "columns": [
        "tosses",
        "mean_absolute_count_error",
        "mean_absolute_proportion_error",
        "count_standard_deviation",
        "proportion_standard_deviation"
      ],
      "rows": [
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          20,
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        ],
        [
          100,
          4.012,
          0.040119999999999996,
          5.0,
          0.05
        ],
        [
          1000,
          12.63405,
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          15.811388300841896,
          0.015811388300841896
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        [
          10000,
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          50.0,
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      ],
      "sha256": "848d08f460183469f29ec9ad040c8e54b8ff7840e71b25cf19f3a1b93d5d2bcf"
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    {
      "id": 11,
      "slug": "modulo-bias",
      "title": "An eight-bit die exposes modulo bias",
      "seed": 20260933,
      "summary": {
        "accepted_per_method": 1000000,
        "rejection_proposals": 1016045,
        "rejection_rate": 0.01579162340250678
      },
      "method": "Map one million uniform bytes to six faces with modulo; separately generate one million accepted faces by rejecting bytes 252\u2013255. Enumerate the 256 byte values to obtain exact probabilities.",
      "limits": "Eight bits deliberately magnify the bias. This is a mapping demonstration using a reproducible PRNG, not a test of Web Crypto or an estimate of bias in a specific lottery.",
      "csv": "modulo-bias.csv",
      "columns": [
        "face",
        "modulo_count",
        "rejection_count",
        "exact_modulo_probability",
        "exact_rejection_probability"
      ],
      "rows": [
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      "sha256": "8b715aea024eda31b7da7642909994aba1fe81ab4eaf395a79fcc1846793a77f"
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}