ᱵᱞᱮᱡᱽ ᱯᱟᱥᱠᱟᱞ: ᱨᱩᱞᱮᱴ ᱪᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱠᱟᱹᱦᱱᱤ
ᱚᱱᱟ ᱜᱚᱱᱤᱛ ᱵᱤᱫᱽ ᱡᱟᱦᱟᱸᱭ ᱥᱟᱫᱷᱟᱨᱚᱱ ᱛᱮ ᱪᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱡᱚᱥ ᱧᱟᱢᱮᱫᱟᱭ, ᱟᱨ ᱡᱟᱦᱟᱸᱭ ᱮᱴᱟᱜ ᱡᱤᱱᱤᱥ ᱛᱮᱭᱟᱨ ᱨᱮ ᱢᱮᱱᱟᱭ ᱛᱟᱦᱮᱸᱫ᱾
ᱯᱟᱥᱠᱟᱞ (1623–1662) 1650 ᱫᱚᱥᱚᱠ ᱨᱮ ᱥᱟᱵᱟᱫᱽ ᱪᱟᱞᱟᱜ ᱠᱟᱞ ᱥᱟᱶ ᱯᱚᱨᱚᱠᱷ ᱠᱮᱫᱟᱭ᱾ ᱩᱱᱤ ᱥᱮᱫ ᱞᱟᱜᱟᱣᱚᱜ ᱠᱟᱱ ᱡᱷᱟᱹᱯᱤ ᱵᱟᱝ ᱜᱷᱟᱥᱚᱜ ᱜᱷᱩᱨᱟᱹᱣ ᱪᱟᱠᱟ ᱫᱚ ᱨᱩᱞᱮᱴ ᱨᱚᱴᱚᱨ ᱨᱮᱭᱟᱜ ᱢᱤᱫ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ ᱥᱮᱫᱟᱭ ᱨᱩᱯ ᱠᱟᱱᱟ; ᱱᱚᱶᱟ ᱡᱚᱥ ᱫᱚ ᱞᱟᱠᱪᱟᱨ ᱜᱮ, ᱠᱟᱜᱚᱡ ᱨᱮ ᱥᱟᱵᱩᱫ ᱵᱟᱹᱱᱩᱜ-ᱟ᱾
ᱜᱚᱱᱤᱛ ᱵᱤᱫᱽ
ᱵᱞᱮᱡᱽ ᱯᱟᱥᱠᱟᱞ (Blaise Pascal) ᱮᱞᱮ ᱥᱮᱨᱢᱟ ᱩᱢᱮᱨ ᱨᱮ ᱢᱤᱫ ᱢᱮᱥᱤᱱ ᱠᱮᱞᱠᱩᱞᱮᱴᱚᱨ ᱛᱮᱭᱟᱨ ᱠᱮᱫᱟᱭ, ᱜᱟᱫᱮᱞ ᱩᱢᱮᱨ ᱨᱮ ᱯᱨᱚᱡᱮᱠᱴᱤᱵᱷ ᱡᱤᱭᱚᱢᱮᱴᱨᱤ ᱨᱮ ᱯᱷᱚᱞ ᱠᱚ ᱥᱟᱵᱩᱫ ᱠᱮᱫᱟᱭ, ᱫᱟᱵ ᱟᱨ ᱠᱷᱟᱹᱞᱤ ᱴᱷᱟᱶ ᱵᱟᱵᱚᱫ ᱵᱩᱱᱤᱭᱟᱫ ᱠᱟᱹᱢᱤ ᱠᱮᱫᱟᱭ, ᱟᱨ 1654 ᱨᱮ ᱯᱤᱭᱮᱨ ᱫᱮ ᱯᱷᱮᱨᱢᱟ (Pierre de Fermat) ᱥᱟᱶ ᱪᱤᱴᱷᱤ ᱛᱮ “ᱯᱚᱭᱮᱱᱴ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱥᱭᱟ” ᱥᱚᱞᱦᱮ ᱠᱮᱫᱟᱭ: ᱛᱟᱞᱟ ᱨᱮ ᱛᱟᱹᱴᱠᱟᱹᱣ ᱦᱩᱭ ᱠᱟᱱ ᱵᱷᱟᱹᱜᱭᱚ ᱠᱷᱮᱞ ᱨᱮᱭᱟᱜ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱪᱮᱫ ᱞᱮᱠᱟ ᱱᱤᱭᱟᱭ ᱛᱮ ᱦᱟᱹᱴᱤᱧ ᱦᱩᱭᱩᱜ-ᱟ᱾ ᱩᱱᱠᱤᱱᱟᱜ ᱛᱮᱞᱟ, ᱠᱷᱮᱞ ᱪᱟᱞᱟᱜ ᱠᱷᱟᱱ ᱡᱚᱛᱚ ᱠᱷᱮᱞᱚᱸᱰ ᱨᱮᱭᱟᱜ ᱡᱤᱛ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱵᱷᱟᱵᱱᱟᱜᱚᱱᱤᱛᱥᱚᱢᱵᱷᱟᱵᱱᱟᱢᱤᱫ ᱨᱮᱡᱟᱞᱴ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱵᱷᱟᱵᱱᱟ, 0 ᱠᱷᱚᱱ 1 ᱟᱨᱵᱟᱝ 0% ᱠᱷᱚᱱ 100%᱾ ᱢᱤᱫ ᱮᱞ: 1/37 = 2.70%᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱞᱮᱠᱟᱛᱮ ᱦᱟᱹᱴᱤᱧ, ᱚᱱᱟ ᱜᱮ ᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝᱜᱚᱱᱤᱛᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝᱡᱮᱞᱮᱧ ᱚᱠᱛᱚ ᱨᱮ ᱢᱤᱫ ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ ᱟᱣᱥᱚᱛ ᱨᱮᱡᱟᱞᱴ᱾ ᱡᱚᱛᱚ ᱥᱟᱫᱷᱟᱨᱚᱱ ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱵᱟᱡᱤ ᱨᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ −1/37᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱮᱭᱟᱜ ᱡᱚᱱᱚᱢ ᱠᱟᱱᱟ᱾ ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ EV ᱦᱤᱥᱟᱹᱵ ᱚᱱᱟ ᱪᱤᱴᱷᱤ ᱠᱚ ᱠᱷᱚᱱ ᱦᱮᱡ ᱟᱠᱟᱱᱟ᱾
ᱵᱞᱮᱡᱽ ᱯᱟᱥᱠᱟᱞ ᱨᱮᱭᱟᱜ ᱨᱩᱞᱮᱴ ᱢᱮᱥᱤᱱ ᱢᱮᱱᱟᱜ ᱛᱟᱦᱮᱸᱫ ᱥᱮ?
1650 ᱫᱚᱥᱚᱠ ᱛᱟᱞᱟ ᱨᱮ ᱯᱟᱥᱠᱟᱞ ᱢᱤᱫ ᱥᱟᱵᱟᱫᱽ ᱪᱟᱞᱟᱜ ᱢᱮᱥᱤᱱ ᱨᱮ ᱠᱟᱹᱢᱤ ᱠᱮᱫᱟᱭ, ᱟᱨ ᱢᱮᱱᱟᱜ-ᱟ ᱡᱮ ᱩᱱᱤ ᱢᱤᱫ ᱥᱟᱦᱤ ᱛᱩᱞᱟᱹ ᱨᱮᱭᱟᱜ ᱪᱟᱠᱟ ᱛᱮᱭᱟᱨ ᱠᱮᱫᱟᱭ ᱡᱟᱦᱟᱸ ᱫᱚ ᱢᱤᱫᱴᱮᱱ ᱜᱷᱟᱥᱚᱜ ᱥᱟᱶ ᱡᱮᱞᱮᱧ ᱚᱠᱛᱚ ᱦᱟᱹᱵᱤᱡ ᱜᱷᱩᱨᱟᱹᱣᱚᱜ ᱛᱟᱦᱮᱸᱫ᱾ ᱱᱚᱶᱟ ᱪᱟᱠᱟ ᱨᱩᱞᱮᱴ ᱨᱚᱴᱚᱨᱥᱟᱡ ᱥᱟᱢᱟᱱᱨᱚᱴᱚᱨᱪᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱜᱷᱩᱨᱟᱹᱣᱚᱜ ᱦᱟᱹᱴᱤᱧ ᱡᱟᱦᱟᱸ ᱨᱮ ᱠᱷᱟᱱᱟ ᱠᱚ ᱢᱮᱱᱟᱜ-ᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱦᱩᱭ ᱮᱱᱟ, ᱱᱚᱶᱟ ᱠᱟᱹᱦᱱᱤ ᱟᱹᱰᱤ ᱤᱛᱤᱦᱟᱹᱥ ᱯᱚᱛᱷᱤ ᱨᱮ ᱫᱩᱦᱲᱟᱹᱣ ᱟᱠᱟᱱᱟ, ᱢᱮᱱᱠᱷᱟᱱ ᱢᱩᱞ ᱯᱷᱮᱰᱟᱛ ᱨᱮ ᱢᱤᱫᱴᱟᱝ ᱨᱮ ᱦᱚᱸ ᱚᱞ ᱵᱟᱹᱱᱩᱜ-ᱟ᱾ 18 ᱥᱟᱭ ᱥᱮᱨᱢᱟ ᱨᱮ ᱥᱯᱟ ᱥᱟᱦᱟᱨ ᱨᱮᱭᱟᱜ ᱪᱟᱞᱟᱣᱤᱭᱟᱹ ᱠᱚ ᱱᱚᱶᱟ ᱧᱩᱛᱩᱢ ᱠᱚ ᱵᱮᱵᱷᱟᱨ ᱞᱮᱫᱟ ᱵᱮᱯᱟᱨ ᱞᱟᱹᱜᱤᱫ, ᱱᱚᱶᱟ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ᱾ ᱞᱮᱵ ᱫᱚ ᱱᱚᱶᱟ ᱥᱟᱵᱩᱫ ᱞᱮᱠᱟᱛᱮ ᱵᱟᱭ ᱟᱠᱟᱨᱤᱭᱟ᱾
ᱥᱚᱢᱵᱷᱟᱵᱱᱟ, ᱩᱱᱤᱭᱟᱜ ᱟᱨ ᱟᱵᱚᱣᱟᱜ ᱠᱟᱛᱷᱟ ᱛᱮ
1654 ᱨᱮᱭᱟᱜ ᱪᱤᱴᱷᱤ ᱠᱚ ᱨᱮ ᱨᱩᱞᱮᱴ ᱨᱮᱭᱟᱜ ᱠᱟᱛᱷᱟ ᱵᱟᱹᱱᱩᱜ-ᱟ, ᱪᱮᱫᱟᱜ ᱡᱮ ᱚᱱᱟ ᱫᱚ ᱵᱟᱝ ᱛᱟᱦᱮᱸᱫ ᱛᱟᱦᱮᱸᱫ᱾ ᱩᱱᱠᱤᱱ ᱠᱩᱞᱤ ᱮᱫᱟ: ᱵᱟᱨ ᱠᱷᱮᱞᱚᱸᱰ ᱠᱤᱱ ᱫᱚ ᱢᱤᱫ ᱯᱚᱴ ᱡᱤᱛᱟᱹᱣ ᱞᱟᱹᱜᱤᱫ ᱯᱮ ᱯᱚᱭᱮᱱᱴ ᱞᱟᱹᱠᱛᱤᱭᱟ; ᱩᱱᱠᱤᱱ 2–1 ᱨᱮ ᱛᱤᱝᱜᱩ ᱠᱤᱱᱟ; ᱯᱚᱴ ᱪᱮᱫ ᱞᱮᱠᱟ ᱦᱟᱹᱴᱤᱧ ᱦᱩᱭᱩᱜ-ᱟ? ᱯᱟᱥᱠᱟᱞ ᱟᱜ ᱛᱮᱞᱟ ᱨᱮ ᱡᱟᱦᱟᱸ ᱵᱮᱵᱷᱟᱨᱚᱜ ᱠᱟᱱᱟ ᱚᱱᱟ ᱱᱤᱛ ᱫᱚ ᱥᱚᱨᱛᱚ ᱟᱸᱫᱟᱡᱽ ᱢᱮᱱᱟ, ᱟᱨ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ ᱞᱟᱦᱟ ᱛᱮᱭᱟᱜ ᱫᱟᱨᱮ ᱞᱟᱛᱟᱨ ᱥᱮᱫ ᱫᱩᱦᱲᱟᱹ ᱫᱩᱦᱲᱟᱹ ᱜᱟᱞᱚᱪ᱾ ᱦᱟᱭᱜᱮᱱᱥ ᱚᱱᱟ ᱪᱤᱴᱷᱤ ᱠᱚ ᱯᱟᱲᱦᱟᱣ ᱠᱮᱫᱟᱭ ᱟᱨ 1657 ᱨᱮ ᱟᱸᱫᱟᱡᱽ ᱵᱟᱵᱚᱫ ᱯᱟᱦᱤᱞ ᱯᱟᱲᱦᱟᱣ ᱯᱚᱛᱷᱤ ᱪᱷᱟᱯᱟᱣ ᱠᱮᱫᱟᱭ᱾ “ᱟᱸᱫᱟᱡᱽ” (expectation) ᱥᱟᱵᱫᱚ ᱫᱚ ᱩᱱᱤᱭᱟᱜ ᱠᱟᱱᱟ᱾
ᱯᱟᱥᱠᱟᱞ ᱟᱨ ᱯᱷᱮᱨᱢᱟ ᱯᱚᱭᱮᱱᱴ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱥᱭᱟ ᱥᱚᱞᱦᱮ ᱠᱤᱱ ᱮᱫᱟ: ᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱫᱚ ᱢᱤᱫ ᱴᱷᱟᱶ ᱨᱮᱭᱟᱜ ᱱᱤᱭᱟᱭ ᱜᱚᱱᱚᱝ᱾
ᱦᱟᱭᱜᱮᱱᱥ, De Ratiociniis in Ludo Aleae: ᱟᱸᱫᱟᱡᱽ ᱵᱟᱵᱚᱫ ᱯᱟᱦᱤᱞ ᱪᱷᱟᱯᱟ ᱚᱞ᱾
ᱭᱟᱠᱚᱵ ᱵᱮᱨᱱᱩᱞᱤ, Ars Conjectandi: ᱢᱟᱨᱟᱝ ᱮᱞ ᱨᱮᱭᱟᱜ ᱱᱤᱭᱚᱢ᱾
ᱯᱮᱨᱤᱥ ᱨᱮ ᱨᱩᱞᱮᱴ ᱵᱟᱵᱚᱫ ᱚᱞ᱾ ᱡᱚᱛᱚ ᱵᱷᱩᱜᱛᱟᱱ ᱫᱚ 36/n − 1: EV = −1/37 (ᱟᱨᱵᱟᱝ −2/38)᱾
ᱯᱤᱭᱟᱨᱥᱚᱱ ᱟᱜ ᱠᱟᱭ-ᱥᱠᱣᱮᱭᱟᱨ ᱯᱚᱨᱚᱠᱷ, ᱨᱩᱞᱮᱴ ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱩᱫᱩᱜ ᱟᱠᱟᱱ᱾
ᱪᱮᱛᱟᱱ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ, ᱱᱚᱶᱟ ᱥᱟᱭᱤᱴ ᱨᱮ, ᱢᱤᱫ ᱵᱽᱨᱟᱣᱡᱟᱨ ᱨᱮ᱾
ᱵᱟᱡᱤ ᱜᱟᱞᱚᱪ
ᱵᱟᱡᱤ ᱵᱟᱵᱚᱫ ᱯᱟᱥᱠᱟᱞ ᱨᱮᱭᱟᱜ ᱡᱟᱹᱥᱛᱤ ᱧᱩᱛᱩᱢ ᱜᱟᱞᱚᱪ ᱫᱚ ᱨᱩᱞᱮᱴ ᱵᱟᱵᱚᱫ ᱫᱚ ᱵᱟᱝ, ᱵᱚᱝᱜᱟ ᱵᱟᱵᱚᱫ ᱠᱟᱱᱟ: Pensées ᱨᱮᱭᱟᱜ “ᱵᱟᱡᱤ”, ᱡᱟᱦᱟᱸ ᱫᱚ ᱵᱤᱥᱣᱟᱥ ᱫᱚ ᱢᱩᱪᱟᱹᱫ ᱵᱟᱹᱱᱩᱜ ᱵᱷᱩᱜᱛᱟᱱ ᱥᱟᱶ ᱢᱤᱫ ᱵᱟᱡᱤ ᱞᱮᱠᱟ ᱩᱫᱩᱜ ᱮᱫᱟ᱾ ᱱᱚᱶᱟ ᱢᱤᱫ ᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱜᱟᱞᱚᱪ ᱠᱟᱱᱟ, ᱟᱨ ᱱᱚᱶᱟ ᱜᱮ ᱯᱟᱦᱤᱞ ᱫᱷᱟᱣ ᱡᱟᱦᱟᱭ ᱦᱚᱸ ᱡᱟᱦᱟᱱᱟᱜ ᱥᱮᱫ ᱜᱟᱞᱚᱪ ᱞᱟᱹᱜᱤᱫ ᱥᱤᱫᱽᱫᱟᱹᱱᱛ ᱛᱚᱛᱷ ᱵᱮᱵᱷᱟᱨ ᱠᱮᱫᱟᱭ᱾ ᱪᱟᱠᱟ ᱞᱟᱹᱜᱤᱫ ᱯᱟᱥᱠᱟᱞ ᱧᱩᱛᱩᱢ ᱟᱹᱜᱩᱭ ᱠᱟᱱ ᱨᱩᱞᱮᱴ ᱠᱷᱮᱞᱚᱸᱰ ᱠᱚ ᱵᱷᱩᱞ ᱯᱟᱥᱠᱟᱞ ᱠᱚ ᱧᱩᱛᱩᱢ ᱮᱫᱟᱭ; ᱞᱟᱹᱠᱛᱤ ᱯᱟᱥᱠᱟᱞ ᱫᱚ ᱚᱱᱟᱭ ᱠᱟᱱᱟ ᱡᱟᱦᱟᱸᱭ ᱩᱫᱩᱜ ᱠᱮᱫᱟ ᱡᱮ ᱢᱤᱫ ᱡᱩᱣᱟ ᱨᱮᱭᱟᱜ ᱜᱚᱱᱚᱝ ᱪᱮᱫ ᱞᱮᱠᱟ ᱛᱷᱤᱨ ᱦᱩᱭᱩᱜ-ᱟ᱾
ᱩᱞᱴᱟᱹ ᱠᱟᱛᱷᱟ
ᱯᱟᱥᱠᱟᱞ ᱯᱟᱦᱤᱞ ᱨᱩᱞᱮᱴ ᱪᱟᱠᱟ ᱛᱮᱭᱟᱨ ᱠᱮᱫᱟᱭ ᱥᱮ ᱵᱟᱝ, ᱠᱷᱮᱞ ᱫᱚ ᱩᱱᱤᱭᱟᱜ ᱥᱟᱹᱨᱤ ᱠᱟᱹᱢᱤ ᱨᱮᱭᱟᱜ ᱢᱤᱫ ᱥᱟᱯᱷᱟ ᱩᱫᱟᱦᱟᱨᱚᱬ ᱠᱟᱱᱟ: ᱢᱤᱫ ᱥᱟᱡ ᱡᱟᱦᱟᱸ ᱫᱚ ᱚᱱᱟ ᱦᱚᱲ ᱠᱷᱚᱱ ᱦᱮᱡ ᱟᱠᱟᱱᱟ ᱡᱟᱦᱟᱸᱭ ᱥᱟᱵᱩᱫ ᱠᱮᱫᱟ ᱡᱮ ᱪᱮᱫ ᱦᱚᱸ ᱵᱟᱝ ᱮᱢ ᱠᱟᱛᱮ ᱪᱮᱫ ᱦᱚᱸ ᱵᱟᱝ ᱧᱟᱢᱚᱜ-ᱟ (ᱥᱟᱵᱟᱫᱽ ᱪᱟᱞᱟᱜ ᱵᱟᱝ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ; ᱟᱸᱫᱟᱡᱽ ᱵᱟᱭ ᱵᱚᱫᱚᱞᱚᱜ-ᱟ), ᱟᱨ ᱚᱱᱟ ᱥᱟᱡ ᱜᱮ ᱥᱮᱨᱢᱟ ᱨᱮ ᱠᱟᱹᱭᱢᱟ ᱢᱤᱞᱤᱭᱚᱱ ᱠᱷᱮᱞᱚᱸᱰ ᱠᱚ ᱴᱷᱮᱱ ᱚᱱᱟ ᱜᱮ ᱠᱟᱛᱷᱟ ᱩᱫᱩᱜ ᱞᱟᱹᱜᱤᱫ ᱵᱮᱵᱷᱟᱨᱚᱜ ᱠᱟᱱᱟ᱾
ᱠᱩᱠᱞᱤ ᱠᱚ
1650 ᱫᱚᱥᱚᱠ ᱮᱛᱦᱚᱵ ᱨᱮ ᱩᱱᱤ ᱯᱮᱨᱤᱥ ᱨᱮ ᱡᱩᱣᱟᱲᱤ ᱠᱚ ᱛᱟᱞᱟ ᱨᱮ ᱪᱟᱞᱟᱜ ᱦᱤᱡᱩᱜ ᱠᱟᱱᱟᱭ, ᱟᱨ ᱚᱱᱟ ᱛᱮ ᱜᱮ ᱯᱚᱭᱮᱱᱴ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱥᱭᱟ ᱥᱮᱣᱟᱞᱤᱭᱮ ᱫᱮ ᱢᱮᱨᱮ ᱦᱚᱛᱮᱛᱮ ᱩᱱᱤ ᱴᱷᱮᱱ ᱥᱮᱴᱮᱨ ᱮᱱᱟ᱾ ᱩᱱᱤ ᱠᱷᱟᱴᱤ ᱛᱮ ᱴᱟᱠᱟ ᱞᱟᱹᱜᱤᱫ ᱠᱷᱮᱞ ᱞᱮᱫᱟᱭ, ᱱᱚᱶᱟ ᱨᱮᱭᱟᱜ ᱡᱟᱦᱟᱸ ᱚᱞ ᱵᱟᱹᱱᱩᱜ-ᱟ᱾
ᱪᱮᱫᱟᱜ ᱡᱮ ᱱᱚᱶᱟ ᱢᱤᱫ ᱵᱮᱥ ᱠᱟᱹᱦᱱᱤ ᱠᱟᱱᱟ ᱟᱨ 19 ᱥᱟᱭ ᱥᱮᱨᱢᱟ ᱨᱮᱭᱟᱜ ᱠᱮᱥᱤᱱᱳ ᱯᱨᱚᱪᱟᱨᱤᱭᱟᱹ ᱠᱚ ᱱᱚᱶᱟ ᱠᱚ ᱠᱩᱥᱤᱭᱟᱜ ᱛᱟᱦᱮᱸᱫ᱾ ᱱᱚᱶᱟ ᱫᱩᱦᱲᱟᱹᱣ ᱡᱩᱫᱤ, ᱢᱤᱫ ᱯᱷᱩᱴᱱᱚᱴ ᱥᱟᱶ᱾
ᱞᱟᱸᱫᱟ ᱞᱟᱹᱜᱤᱫ ᱠᱚᱢᱤᱠ
ᱩᱱᱤ ᱥᱟᱵᱩᱫ ᱠᱮᱫᱟᱭ ᱡᱮ ᱪᱮᱫ ᱦᱚᱸ ᱵᱟᱝ ᱮᱢ ᱠᱟᱛᱮ ᱪᱮᱫ ᱦᱚᱸ ᱵᱟᱝ ᱧᱟᱢᱚᱜ-ᱟ᱾ ᱛᱟᱭᱚᱢ ᱛᱮ ᱡᱟᱦᱟᱭ ᱚᱱᱟ ᱪᱮᱛᱟᱱ ᱨᱮ ᱢᱤᱫ ᱠᱮᱥᱤᱱᱳ ᱛᱮᱭᱟᱨ ᱠᱮᱫᱟᱭ᱾
ᱯᱷᱮᱰᱟᱛ
- Pascal, B. & Fermat, P. Correspondence on the problem of points.
- Huygens, C. De Ratiociniis in Ludo Aleae.
- Kavanagh, T. M. Dice, Cards, Wheels: A Different History of French Culture.