ᱵᱟᱲᱛᱤ · ᱯᱟᱲᱦᱟᱣ ᱛᱟᱹᱞᱤᱠᱟᱹ · 8 ᱢᱤᱱᱤᱴ

ᱨᱩᱞᱮᱴ ᱯᱚᱛᱚᱵ: ᱯᱟᱲᱦᱟᱣ ᱞᱟᱹᱜᱤᱫ ᱛᱟᱹᱞᱤᱠᱟᱹ

ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱡᱟᱦᱟᱸ ᱨᱩᱞᱮᱴ ᱯᱚᱛᱚᱵ ᱟᱨ ᱚᱱᱚᱞ ᱨᱮ ᱛᱟᱦᱮᱸᱱᱟ, ᱵᱟᱨ ᱜᱮᱞ ᱵᱟᱨ, 1657 ᱨᱮᱭᱟᱜ ᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱵᱟᱵᱚᱛ ᱯᱚᱛᱚᱵ ᱠᱷᱚᱱ 2025 ᱨᱮᱭᱟᱜ ᱯᱟᱲᱦᱟᱣ ᱯᱚᱛᱚᱵ ᱦᱟᱹᱵᱤᱡ, ᱡᱚᱛᱚ ᱥᱟᱶ ᱢᱤᱫ ᱛᱷᱟᱨ ᱡᱮ ᱪᱮᱫᱟᱜ ᱞᱟᱹᱠᱛᱤ᱾

ᱠᱷᱟᱴᱚ ᱛᱮᱞᱟ

ᱜᱚᱱᱤᱛ (ᱦᱟᱭᱜᱮᱱᱥ, ᱵᱮᱨᱱᱩᱞᱤ, ᱦᱟᱞᱢᱚᱥ, ᱯᱷᱮᱞᱟᱨ, ᱛᱷᱚᱨᱯ, ᱮᱯᱥᱴᱟᱭᱤᱱ, ᱮᱛᱤᱭᱟᱨ, ᱟᱨ ᱱᱟᱶᱟ ᱯᱟᱲᱦᱟᱣ ᱯᱚᱛᱚᱵ ᱵᱚᱞᱢᱟᱱ, ᱴᱮᱞᱟᱨ ᱟᱨ ᱦᱩᱱᱟᱪᱮᱠ ᱨᱮᱭᱟᱜ), ᱤᱛᱤᱦᱟᱹᱥ (ᱞᱟᱵᱞᱮ, ᱵᱟᱨᱱᱦᱟᱨᱴ, ᱥᱣᱟᱨᱴᱥ, ᱠᱮᱵᱷᱟᱱᱟ, ᱢᱟᱨᱠᱟᱥ, ᱜᱳᱞᱰᱢᱮᱱ), ᱯᱷᱤᱡᱤᱠᱥ (ᱵᱮᱥ, ᱥᱢᱚᱞ ᱟᱨ ᱴᱥᱮ) ᱟᱨ ᱢᱤᱫᱴᱟᱹᱝ ᱩᱯᱚᱱᱟᱹᱥ ᱡᱟᱦᱟᱸ ᱥᱟᱱᱟᱢ ᱠᱷᱚᱱ ᱵᱮᱥ (ᱫᱚᱥᱛᱚᱭᱮᱣᱥᱠᱤ)᱾ ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱨᱮᱭᱟᱜ ᱥᱟᱱᱟᱢ ᱜᱮ ᱱᱚᱶᱟ ᱛᱟᱠᱟ ᱨᱮ ᱛᱟᱦᱮᱸᱱᱟ᱾

ᱡᱟᱹᱥᱛᱤ ᱢᱟᱨᱮ
ᱦᱟᱭᱜᱮᱱᱥ, 1657
ᱡᱟᱹᱥᱛᱤ ᱱᱟᱶᱟ
ᱦᱩᱱᱟᱪᱮᱠ, 2025
ᱡᱟᱹᱥᱛᱤ ᱵᱮᱥ ᱢᱤᱫᱴᱟᱹᱝ ᱯᱚᱛᱚᱵ
ᱮᱛᱤᱭᱟᱨ, 2010
ᱡᱟᱹᱥᱛᱤ ᱵᱮᱥ ᱩᱯᱚᱱᱟᱹᱥ
ᱡᱩᱣᱟᱲᱤ (The Gambler), 1866

ᱜᱚᱱᱤᱛ

ᱯᱚᱛᱚᱵᱥᱮᱨᱢᱟᱪᱮᱫᱟᱜ ᱯᱟᱲᱦᱟᱣ ᱞᱟᱹᱠᱛᱤ
ᱦᱟᱭᱜᱮᱱᱥ, De Ratiociniis in Ludo Aleae1657ᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱵᱟᱵᱚᱛ ᱯᱟᱦᱤᱞ ᱪᱷᱟᱯᱟ ᱚᱞ; EV ᱚᱠᱟ ᱠᱷᱚᱱ ᱦᱮᱡ ᱟᱠᱟᱱᱟ
ᱵᱮᱨᱱᱩᱞᱤ, Ars Conjectandi1713ᱢᱟᱨᱟᱝ ᱮᱞ ᱨᱮᱭᱟᱜ ᱱᱤᱭᱚᱢ, ᱡᱟᱦᱟᱸ ᱥᱤᱥᱴᱮᱢ ᱠᱚ ᱵᱷᱩᱞ ᱠᱚ ᱵᱩᱡᱷᱟᱹᱣᱟ
ᱦᱟᱞᱢᱚᱥ, “Invariants of certain stochastic transformations”1939ᱯᱟᱦᱤᱞ ᱥᱟᱫᱷᱟᱨᱚᱱ ᱯᱨᱚᱢᱟᱱ ᱡᱮ ᱡᱟᱦᱟᱸᱱ ᱵᱟᱡᱤ ᱥᱤᱥᱴᱮᱢ ᱦᱚᱸ ᱟᱸᱫᱟᱡᱽ ᱨᱮᱡᱟᱞᱴ ᱵᱟᱭ ᱵᱚᱫᱚᱞᱟ
ᱯᱷᱮᱞᱟᱨ, An Introduction to Probability Theory, vol. 11968ᱡᱩᱣᱟᱲᱤᱭᱟᱜ ᱯᱩᱸᱡᱤ ᱢᱩᱪᱟᱹᱫ, ᱵᱮᱵᱩᱡᱷ ᱫᱷᱟᱯ, ᱞᱟᱜᱟᱛᱟᱨ; ᱥᱟᱫᱷᱟᱨᱚᱱ ᱥᱮᱸᱫᱽᱨᱟ ᱯᱚᱛᱚᱵ
ᱛᱷᱚᱨᱯ, “Optimal gambling systems for favorable games”1969ᱵᱟᱡᱤ ᱥᱤᱥᱴᱮᱢ ᱡᱚᱠᱷᱚᱱ ᱜᱚᱲᱚ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ (ᱥᱩᱵᱤᱫᱷᱟ ᱠᱷᱮᱞ ᱨᱮ ᱜᱮ) ᱟᱨ ᱥᱤᱥᱴᱮᱢ-ᱵᱟᱹᱱᱩᱜ ᱨᱮᱡᱟᱞᱴ ᱛᱟᱭᱚᱢ ᱨᱮᱱᱟᱜ ᱥᱚᱨᱛᱚ
ᱛᱷᱚᱨᱯ, The Mathematics of Gambling1984ᱨᱩᱞᱮᱴ ᱠᱚᱢᱯᱤᱭᱩᱴᱟᱨ ᱵᱟᱵᱚᱛ ᱚᱱᱟᱭ ᱵᱮᱱᱟᱣᱤᱡ ᱟᱡ ᱛᱮᱭ ᱞᱟᱹᱭ ᱟᱠᱟᱫᱟ, ᱵᱟᱲᱛᱤ ᱛᱟᱦᱮᱸ ᱠᱷᱟᱱ ᱛᱤᱱᱟᱹᱜ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱞᱟᱹᱠᱛᱤ ᱚᱱᱟ ᱦᱤᱥᱟᱹᱵ ᱥᱟᱶ
ᱮᱯᱥᱴᱟᱭᱤᱱ, The Theory of Gambling and Statistical Logic2013 (2 ᱥᱟᱭ ᱥᱟᱸᱥᱠᱟᱨᱚᱬ)ᱥᱟᱱᱟᱢ ᱠᱮᱥᱤᱱᱳ ᱠᱷᱮᱞ ᱨᱮᱭᱟᱜ ᱡᱟᱸᱪ; ᱚᱱᱟ ᱵᱷᱤᱛᱤᱨ ᱨᱩᱞᱮᱴ
ᱮᱛᱤᱭᱟᱨ, The Doctrine of Chances2010ᱪᱷᱟᱯᱟ ᱨᱮ ᱨᱩᱞᱮᱴ ᱨᱮᱭᱟᱜ ᱡᱟᱹᱥᱛᱤ ᱯᱩᱨᱟᱹ ᱜᱚᱱᱤᱛ ᱵᱤᱪᱟᱹᱨ
ᱵᱚᱞᱢᱟᱱ, Mathematics of the Big Four Casino Table Games2021ᱫᱷᱟᱨᱮ ᱵᱟᱡᱤ, ᱮᱴᱟᱜ ᱞᱮᱠᱟᱱ ᱪᱟᱠᱟ ᱟᱨ ᱛᱟᱥ ᱨᱩᱞᱮᱴ ᱨᱮᱭᱟᱜ ᱫᱟᱢ ᱢᱤᱫ ᱢᱤᱫ ᱛᱮ, ᱠᱟᱹᱢᱤ ᱥᱟᱶ
ᱴᱮᱞᱟᱨ, Games, Gambling, and Probability2021 (2 ᱥᱟᱭ ᱥᱟᱸᱥᱠᱟᱨᱚᱬ)ᱞᱟᱜᱟᱛᱟᱨ, ᱠᱮᱞᱤ ᱵᱟᱡᱤ ᱟᱨ ᱡᱩᱣᱟᱲᱤᱭᱟᱜ ᱯᱩᱸᱡᱤ ᱢᱩᱪᱟᱹᱫ, ᱠᱚᱨᱥ ᱞᱮᱠᱟ
ᱵᱚᱞᱢᱟᱱ, Basic Gambling Mathematics2024 (2 ᱥᱟᱭ ᱥᱟᱸᱥᱠᱟᱨᱚᱬ)ᱠᱮᱥᱤᱱᱳ ᱯᱨᱚᱢᱚᱥᱚᱱ ᱟᱨ ᱨᱤᱵᱮᱴ, ᱟᱨ ᱨᱮᱡᱟᱞᱴ ᱵᱳᱨᱰ ᱠᱮᱥᱤᱱᱳ ᱞᱟᱹᱜᱤᱫ ᱛᱤᱱᱟᱹᱜ ᱞᱚᱜᱚᱱ ᱟᱡᱟᱜ ᱫᱟᱢ ᱨᱩᱣᱟᱹᱲ ᱟᱹᱜᱩᱭᱟ
ᱦᱩᱱᱟᱪᱮᱠ, Quantitative Literacy Through Games and Gambling2025ᱞᱤᱞᱤᱡ ᱯᱟᱦᱤᱞ ᱠᱚᱨᱥ; ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱨᱮ ᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱢᱮᱱᱟᱜ-ᱟ ᱢᱮᱱᱛᱮ ᱚᱱᱟᱜ ᱠᱟᱛᱷᱟ ᱧᱮᱞ ᱯᱮ, ᱡᱟᱦᱟᱸ ᱢᱚᱬᱮ ᱮᱞ ᱵᱟᱡᱤ ᱨᱟᱹᱯᱩᱫᱟ

ᱤᱛᱤᱦᱟᱹᱥ

ᱯᱚᱛᱚᱵᱥᱮᱨᱢᱟᱪᱮᱫᱟᱜ ᱯᱟᱲᱦᱟᱣ ᱞᱟᱹᱠᱛᱤ
ᱞᱟᱵᱞᱮ, La Roulette, ou le Jour1801ᱠᱷᱮᱞ ᱨᱮᱭᱟᱜ ᱯᱟᱦᱤᱞ ᱯᱩᱨᱟᱹ ᱵᱤᱵᱨᱚᱬ, ᱢᱤᱫᱴᱟᱹᱝ ᱩᱯᱚᱱᱟᱹᱥ ᱨᱮ
ᱵᱟᱨᱱᱦᱟᱨᱴ, Beating the Wheel1992ᱵᱟᱭᱟᱥ ᱠᱷᱮᱞ ᱨᱮᱭᱟᱜ ᱤᱛᱤᱦᱟᱹᱥ: ᱡᱮᱜᱟᱨ, ᱯᱮᱞᱟᱭᱳ ᱠᱚ ᱞᱟᱦᱟ ᱨᱮᱱᱟᱜ ᱠᱷᱮᱞᱚᱸᱰ ᱠᱚ, ᱦᱚᱨᱟ ᱠᱚ
ᱥᱣᱟᱨᱴᱥ, Roll the Bones2006ᱡᱩᱣᱟ ᱨᱮᱭᱟᱜ ᱥᱟᱫᱷᱟᱨᱚᱱ ᱤᱛᱤᱦᱟᱹᱥ; ᱚᱱᱟ ᱨᱮ ᱨᱩᱞᱮᱴ ᱨᱮᱭᱟᱜ ᱴᱷᱟᱶ
ᱠᱮᱵᱷᱟᱱᱟ, Dice, Cards, Wheels2005ᱯᱷᱨᱮᱧᱪ ᱡᱩᱣᱟ ᱞᱟᱠᱪᱟᱨ; ᱯᱟᱞᱮ-ᱨᱣᱟᱭᱟᱞ ᱟᱨ ᱥᱯᱟ ᱥᱚᱦᱚᱨ ᱠᱚ
ᱢᱟᱨᱠᱟᱥ, American Roulette2003ᱯᱟᱥᱴ-ᱯᱚᱥᱴᱤᱝ, ᱵᱷᱤᱛᱤᱨ ᱠᱷᱚᱱ ᱵᱤᱵᱨᱚᱬ, ᱚᱱᱟ ᱨᱮᱭᱟᱜ ᱡᱟᱹᱥᱛᱤ ᱧᱩᱛᱩᱢᱟᱱ ᱠᱟᱹᱢᱤᱭᱤᱡ ᱛᱮ
ᱜᱳᱞᱰᱢᱮᱱ, Great Gambling Scams2007ᱱᱤᱥ ᱨᱮ ᱞᱤ, ᱢᱟᱨᱠᱟᱥ ᱟᱨ ᱨᱤᱴᱥ ᱫᱚᱦᱲᱟ ᱞᱟᱹᱭ ᱟᱠᱟᱱ; ᱨᱚᱥ ᱢᱮᱱᱟᱜ-ᱟ, ᱟᱨ ᱟᱭᱢᱟ ᱠᱟᱛᱷᱟ ᱨᱮ ᱵᱷᱩᱞ (ᱧᱮᱞ ᱯᱮ ᱨᱩᱞᱮᱴ ᱨᱮ ᱠᱷᱚᱞ)

ᱯᱷᱤᱡᱤᱠᱥ ᱟᱨ ᱮᱰᱣᱟᱱᱴᱮᱡ ᱯᱞᱮ

ᱯᱚᱛᱚᱵᱥᱮᱨᱢᱟᱪᱮᱫᱟᱜ ᱯᱟᱲᱦᱟᱣ ᱞᱟᱹᱠᱛᱤ
ᱵᱮᱥ, The Eudaemonic Pie1985ᱡᱩᱛᱟ ᱠᱚᱢᱯᱤᱭᱩᱴᱟᱨ ᱨᱮᱭᱟᱜ ᱠᱟᱹᱦᱱᱤ; ᱨᱩᱞᱮᱴ ᱯᱷᱤᱡᱤᱠᱥ ᱵᱟᱵᱚᱛ ᱡᱟᱹᱥᱛᱤ ᱵᱮᱥ ᱥᱟᱫᱷᱟᱨᱚᱱ ᱯᱚᱛᱚᱵ
ᱥᱢᱚᱞ ᱟᱨ ᱴᱥᱮ, “Predicting the outcome of roulette”2012ᱞᱟᱦᱟ ᱠᱟᱛᱷᱟ ᱨᱮᱭᱟᱜ ᱯᱷᱤᱡᱤᱠᱥ, ᱢᱟᱯ ᱠᱟᱛᱮ; ᱵᱟᱲᱛᱤ ᱞᱟᱹᱜᱤᱫ ᱪᱮᱫ ᱞᱟᱹᱠᱛᱤ
ᱯᱤᱭᱟᱨᱥᱚᱱ, “On the criterion…”1900ᱠᱟᱭ-ᱥᱠᱣᱮᱭᱟᱨ ᱯᱚᱨᱚᱠᱷ, ᱨᱩᱞᱮᱴ ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱩᱫᱩᱜ ᱠᱟᱛᱮ

ᱨᱩᱞᱮᱴ ᱪᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱯᱷᱤᱡᱤᱠᱥ

ᱩᱯᱚᱱᱟᱹᱥ

ᱫᱚᱥᱛᱚᱭᱮᱣᱥᱠᱤ (Dostoevsky), ᱡᱩᱣᱟᱲᱤ (The Gambler, 1866)᱾ ᱪᱷᱟᱯᱟᱣᱤᱡ ᱥᱟᱶ ᱠᱚᱱᱴᱨᱟᱠᱴ ᱨᱮᱭᱟᱜ ᱢᱩᱪᱟᱹᱫ ᱢᱟᱦᱟᱸ ᱯᱩᱨᱟᱹᱣ ᱞᱟᱹᱜᱤᱫ 26 ᱫᱤᱱ ᱨᱮ ᱚᱞ ᱟᱠᱟᱱ, ᱨᱩᱞᱮᱴ ᱛᱮ ᱟᱹᱰᱤ ᱨᱤᱱ ᱨᱮ ᱛᱟᱦᱮᱸᱱ ᱚᱞᱤᱭᱟᱹ ᱛᱮ; ᱡᱩᱣᱟᱲᱤᱭᱟᱜ ᱵᱷᱩᱞ, ᱥᱤᱥᱴᱮᱢ, ᱵᱟᱝ ᱛᱤᱝᱜᱩ ᱫᱟᱲᱮᱭᱟᱜ ᱞᱟᱞᱚᱥ ᱟᱨ ᱥᱯᱟ ᱥᱚᱦᱚᱨ ᱨᱮᱭᱟᱜ ᱠᱮᱥᱤᱱᱳ, ᱵᱷᱤᱛᱤᱨ ᱠᱷᱚᱱ᱾ ᱪᱮᱛᱟᱱ ᱨᱮᱭᱟᱜ ᱡᱟᱦᱟᱸᱱ ᱨᱩᱞᱮᱴ ᱯᱚᱛᱚᱵ ᱠᱷᱚᱱ ᱠᱷᱟᱴᱚ ᱟᱨ ᱡᱟᱹᱥᱛᱤ ᱠᱷᱚᱱ ᱵᱮᱥᱤ ᱥᱟᱹᱨᱤ᱾

ᱵᱟᱰ ᱦᱚᱢᱵᱩᱨᱜ ᱟᱨ ᱥᱯᱟ ᱠᱮᱥᱤᱱᱳ ᱠᱚᱫᱚᱥᱛᱚᱭᱮᱵᱷᱥᱠᱤ ᱟᱜ ᱨᱩᱞᱮᱴᱮᱱᱵᱩᱨᱜ

ᱡᱚᱛᱚ ᱡᱚᱨᱩᱨ ᱞᱟᱹᱜᱤᱫ ᱡᱟᱹᱥᱛᱤ ᱵᱮᱥ ᱨᱩᱞᱮᱴ ᱯᱚᱛᱚᱵ

ᱦᱤᱥᱟᱹᱵ ᱞᱟᱹᱜᱤᱫ, ᱮᱛᱤᱭᱟᱨ (Ethier) ᱟᱨ ᱛᱟᱭᱚᱢ ᱯᱷᱮᱞᱟᱨ (Feller)᱾ ᱠᱟᱹᱦᱱᱤ ᱞᱟᱹᱜᱤᱫ, ᱵᱟᱨᱱᱦᱟᱨᱴ (Barnhart) ᱟᱨ ᱵᱮᱥ (Bass)᱾ ᱦᱚᱲ ᱠᱚ ᱪᱮᱫᱟᱜ ᱵᱟᱨᱚᱡ ᱦᱚᱸ ᱠᱚ ᱠᱷᱮᱞᱟ, ᱚᱱᱟ ᱞᱟᱹᱜᱤᱫ ᱫᱚᱥᱛᱚᱭᱮᱣᱥᱠᱤ᱾ ᱡᱚᱛᱚ ᱥᱤᱥᱴᱮᱢ ᱛᱟᱨᱠᱚ ᱢᱩᱪᱟᱹᱫ ᱩᱫᱩᱜ ᱯᱨᱚᱢᱟᱱ ᱞᱟᱹᱜᱤᱫ, ᱦᱟᱞᱢᱚᱥ (Halmos), ᱟᱨ ᱚᱱᱟ ᱜᱮ ᱨᱮᱡᱟᱞᱴ ᱥᱚᱦᱡ ᱟᱹᱲᱟᱹ ᱛᱮ ᱞᱟᱹᱜᱤᱫ, ᱛᱷᱚᱨᱯ (Thorp) ᱨᱮᱭᱟᱜ 1984 ᱯᱚᱛᱚᱵ᱾ ᱠᱟᱹᱢᱤ ᱞᱟᱹᱜᱤᱫ, ᱴᱮᱞᱟᱨ (Taylor) ᱥᱮ ᱵᱚᱞᱢᱟᱱ (Bollman)᱾ ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ ᱚᱱᱚᱞ ᱨᱮᱭᱟᱜ ᱯᱷᱮᱰᱟᱛ ᱵᱟᱠᱥᱚ ᱫᱚ ᱱᱚᱶᱟ ᱛᱟᱠᱟ ᱠᱷᱚᱱ ᱜᱮ ᱫᱟᱹᱭᱠᱟᱹ ᱮᱢᱟ᱾

ᱠᱩᱠᱞᱤ ᱠᱚ

ᱵᱮᱥ ᱨᱩᱞᱮᱴ ᱥᱴᱨᱮᱴᱮᱡᱤ ᱯᱚᱛᱚᱵ ᱢᱮᱱᱟᱜ-ᱟ ᱥᱮ?

ᱡᱟᱦᱟᱸ ᱥᱤᱥᱴᱮᱢ ᱠᱟᱹᱵᱤᱡ ᱠᱚᱨᱟᱣᱟ, ᱚᱱᱟ ᱞᱮᱠᱟᱱ ᱵᱮᱥ ᱵᱟᱹᱱᱩᱜ-ᱟ᱾ ᱛᱷᱚᱨᱯ ᱨᱮᱭᱟᱜ 1984 ᱯᱚᱛᱚᱵ ᱨᱮ ᱩᱱᱠᱩ ᱪᱮᱫᱟᱜ ᱯᱷᱮᱞ ᱠᱚ ᱦᱩᱭᱩᱜ-ᱟ ᱚᱱᱟ ᱵᱟᱵᱚᱛ ᱢᱤᱫᱴᱟᱹᱝ ᱠᱷᱟᱴᱚ ᱪᱮᱯᱴᱟᱨ ᱢᱮᱱᱟᱜ-ᱟ; ᱮᱯᱥᱴᱟᱭᱤᱱ ᱟᱨ ᱮᱛᱤᱭᱟᱨ ᱡᱚᱛᱚ ᱥᱤᱥᱴᱮᱢ ᱨᱮᱭᱟᱜ ᱨᱩᱯ ᱵᱩᱡᱷᱟᱹᱣ ᱮᱫᱟ᱾

ᱯᱤᱭᱟᱨᱥᱚᱱ ᱥᱮ ᱦᱟᱭᱜᱮᱱᱥ ᱚᱠᱟ ᱨᱮ ᱯᱟᱲᱦᱟᱣ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ?

ᱵᱟᱨᱭᱟ ᱜᱮ ᱠᱚᱯᱤᱨᱟᱭᱤᱴ ᱠᱷᱚᱱ ᱵᱟᱦᱨᱮ ᱟᱨ ᱚᱱᱞᱟᱭᱤᱱ ᱢᱮᱱᱟᱜ-ᱟ᱾ ᱯᱤᱭᱟᱨᱥᱚᱱ ᱫᱚ ᱠᱟᱴᱷᱤᱱ; ᱦᱟᱭᱜᱮᱱᱥ ᱫᱚ ᱠᱷᱟᱴᱚ ᱟᱨ ᱛᱚᱨᱡᱚᱢᱟ ᱨᱮ ᱟᱹᱰᱤ ᱟᱞᱜᱟ ᱛᱮ ᱯᱟᱲᱦᱟᱣ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ᱾

ᱞᱟᱸᱫᱟ ᱞᱟᱹᱜᱤᱫ ᱠᱚᱢᱤᱠ

ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱨᱮᱭᱟᱜ ᱥᱟᱱᱟᱢ ᱜᱮ ᱱᱚᱶᱟ ᱛᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱞᱟᱛᱟᱨ ᱴᱤᱯᱩᱱᱤ ᱠᱟᱱᱟ᱾

ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱨᱮᱭᱟᱜ ᱥᱟᱱᱟᱢ ᱜᱮ ᱱᱚᱶᱟ ᱛᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱞᱟᱛᱟᱨ ᱴᱤᱯᱩᱱᱤ ᱠᱟᱱᱟ᱾

ᱯᱷᱮᱰᱟᱛ

  1. Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010᱾ ᱪᱮᱯᱴᱟᱨ 13, ᱨᱩᱞᱮᱴ᱾
  2. Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, 3rd ed., 1968᱾ ᱡᱩᱣᱟᱲᱤᱭᱟᱜ ᱯᱩᱸᱡᱤ ᱢᱩᱪᱟᱹᱫ, ᱪᱮᱯᱴᱟᱨ XIV᱾
  3. Halmos, P. R. “Invariants of certain stochastic transformations: the mathematical theory of gambling systems.” Duke Mathematical Journal 5(2), 1939.
  4. Thorp, E. O. “Optimal gambling systems for favorable games.” Review of the International Statistical Institute 37(3), 1969.
  5. Thorp, E. O. The Mathematics of Gambling. Gambling Times / Lyle Stuart, 1984.
  6. Bass, T. A. The Eudaemonic Pie. Houghton Mifflin, 1985.
  7. Barnhart, R. T. Beating the Wheel. Lyle Stuart, 1992.
  8. Bollman, M. Mathematics of the Big Four Casino Table Games: Blackjack, Baccarat, Craps, & Roulette. CRC Press, 2021᱾ ᱪᱮᱯᱴᱟᱨ 2, ᱨᱩᱞᱮᱴ᱾
  9. Bollman, M. Basic Gambling Mathematics: The Numbers Behind the Neon, 2nd ed. CRC Press, 2024.
  10. Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed. CRC Press, 2021.
  11. Hunacek, M. Quantitative Literacy Through Games and Gambling. Chapman & Hall/CRC, 2025.
  12. Marcus, R. American Roulette: How I Turned the Odds Upside Down. St. Martin’s Press, 2003.
  13. Goldman, N. Great Gambling Scams. John Blake, 2007.
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