ᱨᱩᱞᱮᱴ ᱜᱚᱱᱤᱛ · ᱜᱚᱱᱤᱛ ᱥᱚᱢᱥᱭᱟ · 10 ᱢᱤᱱᱤᱴ

ᱥᱚᱢᱥᱭᱟ: ᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹ

$500 ᱴᱮᱵᱩᱞ ᱨᱮ $10 ᱢᱟᱨᱴᱤᱝᱜᱮᱞ᱾ ᱱᱚᱶᱟ ᱛᱤᱱᱟᱹᱜ ᱫᱷᱟᱣ ᱨᱟᱯᱩᱫᱚᱜ-ᱟ, ᱨᱟᱯᱩᱫ ᱚᱠᱛᱚ ᱛᱤᱱᱟᱹᱜ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ, ᱟᱨ ᱥᱤᱢᱟᱹ ᱪᱮᱛᱟᱱ ᱛᱮ ᱟᱹᱜᱩ ᱫᱚ ᱜᱚᱲᱚᱭᱟ ᱥᱮ?

ᱠᱷᱟᱴᱚ ᱛᱮᱞᱟ

ᱞᱟᱜᱟᱛᱟᱨ ᱛᱩᱨᱩᱭ ᱦᱟᱹᱱᱤ ᱛᱮ ᱱᱚᱶᱟ ᱨᱟᱯᱩᱫᱚᱜ-ᱟ: ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ 1.8%, 100 ᱜᱷᱩᱨᱟᱹᱣ ᱥᱮᱥᱚᱱ ᱨᱮᱭᱟᱜ ᱠᱚᱢ ᱵᱮᱥᱤ 60%᱾ ᱨᱟᱯᱩᱫ ᱨᱮᱭᱟᱜ ᱠᱷᱚᱨᱚᱪ $630᱾ ᱥᱤᱢᱟᱹ $5,000 ᱦᱟᱹᱵᱤᱡ ᱪᱮᱛᱟᱱ ᱛᱮ ᱟᱹᱜᱩ ᱞᱮᱠᱷᱟᱱ ᱨᱟᱯᱩᱫ ᱫᱚ ᱠᱚᱢ ᱦᱩᱭᱩᱜ-ᱟ (ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ 0.25%) ᱟᱨ ᱢᱟᱨᱟᱝ ($5,110)᱾ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱟᱠᱟᱱ ᱡᱚᱛᱚ ᱰᱚᱞᱟᱨ ᱨᱮ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤ ᱫᱚ ᱡᱟᱦᱟᱸ ᱥᱤᱢᱟᱹ ᱨᱮ ᱦᱚᱸ 2.70%᱾

ᱮᱦᱚᱵ ᱵᱟᱡᱤ / ᱥᱤᱢᱟᱹ
$10 / $500
ᱨᱟᱯᱩᱫ ᱦᱟᱹᱵᱤᱡ ᱦᱟᱹᱱᱤ
6
ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ p(ᱨᱟᱯᱩᱫ)
1.8%
ᱨᱟᱯᱩᱫ ᱨᱮ ᱦᱟᱹᱱᱤ
$630

ᱥᱚᱢᱥᱭᱟ

ᱢᱤᱫ ᱠᱷᱮᱞᱚᱸᱰ ᱟᱨᱟᱜᱵᱟᱡᱤᱟᱨᱟᱜ18 ᱟᱨᱟᱜ ᱮᱞ ᱨᱮ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ᱾ ᱯᱷᱨᱮᱧᱪ ᱨᱮ: ᱨᱩᱡᱽ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱮ $10 ᱵᱟᱡᱤ ᱮ ᱞᱟᱜᱟᱣᱟ ᱟᱨ ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱵᱟᱨ ᱫᱷᱟᱣ ᱮ ᱵᱮᱱᱟᱣᱟ, ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱫᱚᱦᱲᱟ ᱮᱦᱚᱵ ᱠᱟᱛᱮ᱾ ᱴᱮᱵᱩᱞ ᱨᱮᱭᱟᱜ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱫᱚ $500᱾ (a) ᱛᱤᱱᱟᱹᱜ ᱞᱟᱜᱟᱛᱟᱨ ᱦᱟᱹᱱᱤ ᱛᱮ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱᱥᱴᱨᱮᱴᱮᱡᱤᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱᱢᱟᱨᱮ ᱨᱮᱡᱟᱞᱴ ᱞᱮᱠᱟ ᱛᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱵᱚᱫᱚᱞ ᱡᱟᱦᱟᱸᱱ ᱱᱤᱭᱚᱢ᱾ ᱱᱮᱜᱮᱴᱤᱵᱷ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱵᱟᱲᱦᱟᱣᱟ, ᱯᱚᱡᱤᱴᱤᱵᱷ ᱡᱤᱛ ᱛᱟᱭᱚᱢ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱟᱯᱩᱫᱚᱜ-ᱟ? (b) ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ ᱥᱚᱢᱵᱷᱟᱵᱱᱟᱜᱚᱱᱤᱛᱥᱚᱢᱵᱷᱟᱵᱱᱟᱢᱤᱫ ᱨᱮᱡᱟᱞᱴ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱵᱷᱟᱵᱱᱟ, 0 ᱠᱷᱚᱱ 1 ᱟᱨᱵᱟᱝ 0% ᱠᱷᱚᱱ 100%᱾ ᱢᱤᱫ ᱮᱞ: 1/37 = 2.70%᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱛᱤᱱᱟᱹᱜ? (c) ᱨᱟᱯᱩᱫ ᱨᱮ ᱛᱤᱱᱟᱹᱜ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ? (d) $5,000 ᱥᱤᱢᱟᱹ ᱛᱮ ᱛᱮᱞᱟ ᱠᱚ ᱪᱮᱫ ᱞᱮᱠᱟ ᱵᱚᱫᱚᱞᱚᱜ-ᱟ? (e) ᱡᱟᱦᱟᱱᱟᱜ ᱥᱤᱢᱟᱹ ᱫᱚ ᱥᱤᱥᱴᱮᱢ ᱞᱟᱵᱷ ᱨᱮ ᱟᱹᱜᱩᱭᱟ ᱥᱮ?

ᱨᱩᱞᱮᱴ ᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹ

ᱥᱟᱢᱵᱷᱟᱲ (a)–(c)

ᱵᱟᱡᱤ ᱴᱟᱠᱟ: 10, 20, 40, 80, 160, 320᱾ ᱮᱭᱟᱭ ᱛᱮᱭᱟᱜ ᱫᱚ 640 > 500 ᱦᱩᱭᱩᱜ-ᱟ, ᱚᱱᱟᱛᱮ ᱛᱩᱨᱩᱭ ᱦᱟᱹᱱᱤ ᱛᱮ ᱨᱟᱯᱩᱫᱚᱜ-ᱟ᱾ (b) ᱟᱨᱟᱜ ᱨᱮ p(ᱦᱟᱹᱱᱤ) = 19/37; (19/37)⁶ = 0.0183, ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ ᱠᱚᱢ ᱵᱮᱥᱤ 1.8%᱾ ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱫᱚ ᱟᱣᱥᱚᱛ ᱛᱮ ᱠᱚᱢ ᱵᱮᱥᱤ ᱵᱟᱨ ᱜᱷᱩᱨᱟᱹᱣ ᱛᱟᱦᱮᱸᱱᱟ, ᱚᱱᱟᱛᱮ 100 ᱜᱷᱩᱨᱟᱹᱣ ᱥᱮᱥᱚᱱ ᱨᱮ ᱠᱚᱢ ᱵᱮᱥᱤ 50 ᱪᱟᱠᱨᱟ ᱟᱨ p(ᱠᱚᱢ ᱥᱮ ᱠᱚᱢ ᱢᱤᱫ ᱨᱟᱯᱩᱫ) ≈ 1 − 0.9817⁵⁰ ≈ 60%᱾ (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ᱡᱟᱦᱟᱸ ᱫᱚ 63 ᱮᱦᱚᱵ ᱭᱩᱱᱤᱴᱴᱟᱠᱟᱭᱩᱱᱤᱴᱢᱤᱫ ᱥᱤᱥᱴᱮᱢ ᱟᱨᱵᱟᱝ ᱠᱷᱮᱞᱚᱸᱰ ᱵᱮᱵᱷᱟᱨ ᱵᱮᱥ ᱵᱟᱡᱤ ᱴᱟᱠᱟ᱾ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱠᱚ ᱭᱩᱱᱤᱴ ᱛᱮ ᱵᱩᱡᱷᱟᱹᱣᱚᱜ-ᱟ ᱡᱮᱢᱚᱱ ᱡᱟᱦᱟᱸᱱ ᱴᱮᱵᱩᱞ ᱨᱮ ᱞᱟᱹᱜᱩ ᱦᱩᱭᱩᱜ-ᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ →, ᱪᱟᱠᱨᱟ ᱡᱤᱛᱟᱹᱣ ᱪᱮᱥᱴᱟ ᱮᱫ $10 ᱛᱩᱞᱟᱹ ᱨᱮ᱾

ᱥᱤᱢᱟᱹᱨᱟᱯᱩᱫ ᱦᱟᱹᱵᱤᱡ ᱦᱟᱹᱱᱤᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ p(ᱨᱟᱯᱩᱫ)ᱨᱟᱯᱩᱫ ᱨᱮ ᱦᱟᱹᱱᱤp(100 ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱨᱟᱯᱩᱫ)ᱨᱩᱣᱟᱹᱲ ᱞᱟᱹᱜᱤᱫ ᱞᱟᱹᱠᱛᱤ ᱡᱤᱛ
$10046.95%$150≈ 97%15
$50061.8%$630≈ 60%63
$1,00070.94%$1,270≈ 38%127
$2,00080.48%$2,550≈ 22%255
$5,00090.25%$5,110≈ 11%511
$20,000110.07%$20,470≈ 3%2,047
ᱪᱮᱛᱟᱱ ᱥᱤᱢᱟᱹ ᱵᱟᱨ ᱫᱷᱟᱣ ᱞᱮᱠᱷᱟᱱ ᱨᱟᱯᱩᱫ ᱫᱚ ᱠᱚᱢ ᱵᱮᱥᱤ ᱟᱫᱷᱟ ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱟᱨ ᱵᱟᱨ ᱫᱷᱟᱣ ᱜᱟᱨᱟᱝ ᱦᱩᱭᱩᱜ-ᱟ᱾ ᱜᱩᱬ ᱫᱚ ᱯᱷᱟᱞᱟᱹᱣ ᱜᱮ ᱵᱟᱝ ᱵᱚᱫᱚᱞᱚᱜ-ᱟ᱾

ᱥᱟᱢᱵᱷᱟᱲ (d)

$5,000 ᱨᱮ: 2,560 ᱦᱟᱹᱵᱤᱡ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱦᱩᱠᱩᱢ ᱢᱮᱱᱟᱜ-ᱟ; ᱜᱮᱞ ᱛᱮᱭᱟᱜ ᱵᱟᱡᱤ ᱴᱟᱠᱟ, 5,120, ᱵᱟᱝ ᱦᱟᱛᱟᱣᱚᱜ-ᱟ᱾ ᱟᱨᱮ ᱦᱟᱹᱱᱤ ᱛᱮ ᱨᱟᱯᱩᱫᱚᱜ-ᱟ: (19/37)⁹ ≈ ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ 0.25%, 100 ᱜᱷᱩᱨᱟᱹᱣ ᱥᱮᱥᱚᱱ ᱨᱮᱭᱟᱜ ᱠᱚᱢ ᱵᱮᱥᱤ 11%᱾ ᱨᱟᱯᱩᱫ ᱨᱮ ᱦᱟᱹᱱᱤ: 10 + 20 + … + 2,560 = $5,110᱾ ᱠᱷᱮᱞᱚᱸᱰ ᱫᱚ ᱢᱤᱫ ᱢᱟᱨᱟᱝ ᱵᱤᱯᱚᱫ ᱨᱮᱭᱟᱜ ᱫᱟᱢ ᱛᱮ ᱢᱤᱫ ᱠᱚᱢ ᱦᱩᱭᱩᱜ ᱵᱤᱯᱚᱫ ᱮ ᱠᱤᱨᱤᱧ ᱟᱠᱟᱫᱟ᱾ ᱟᱭᱢᱟ ᱥᱮᱥᱚᱱ ᱨᱮ ᱨᱟᱯᱩᱫ ᱨᱮᱭᱟᱜ ᱟᱸᱫᱟᱡᱽ ᱠᱷᱚᱨᱚᱪ ᱫᱚ p × ᱦᱟᱹᱱᱤ ≈ 0.0183 × 630 = $500 ᱥᱤᱢᱟᱹ ᱨᱮ ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ $11.5 ᱟᱨ 0.0025 × 5,110 = $5,000 ᱨᱮ $12.8; ᱵᱟᱨᱭᱟ ᱫᱚ ᱢᱤᱫ ᱞᱮᱠᱟ ᱧᱮᱞᱚᱜ-ᱟ ᱢᱮᱱᱠᱷᱟᱱ ᱥᱚᱢᱟᱱ ᱫᱚ ᱵᱟᱝ, ᱟᱨ ᱡᱚᱛᱚ ᱫᱚ ᱪᱟᱠᱨᱟ ᱨᱮᱭᱟᱜ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤᱜᱚᱱᱤᱛᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱜᱩᱬ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ; ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱟᱨᱵᱟᱝ ᱚᱠᱛᱚ ᱨᱮ ᱟᱣᱥᱚᱛ ᱛᱮ ᱦᱟᱨᱟᱣᱚᱜ ᱴᱟᱠᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱥᱟᱶ ᱡᱤᱛ ᱪᱟᱠᱨᱟ ᱠᱚᱣᱟᱜ ᱟᱹᱭᱩᱨ $10 ᱠᱟᱱᱟ᱾

ᱥᱟᱢᱵᱷᱟᱲ (e)

ᱵᱟᱝ᱾ ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤ = (ᱪᱟᱠᱨᱟ ᱨᱮᱭᱟᱜ ᱟᱸᱫᱟᱡᱽ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ) / 37, ᱡᱟᱦᱟᱸ ᱥᱤᱢᱟᱹ ᱨᱮ ᱦᱚᱸ, ᱪᱮᱫᱟᱜ ᱥᱮ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱫᱚ ᱟᱡ ᱨᱮᱭᱟᱜ −1/37᱾ ᱥᱤᱢᱟᱹ ᱫᱚ ᱠᱷᱟᱹᱞᱤ ᱴᱷᱟᱹᱣᱠᱟᱹᱣᱟ ᱡᱮ ᱦᱟᱹᱱᱤ ᱫᱚ ᱟᱭᱢᱟ ᱦᱩᱰᱤᱧ ᱡᱤᱛ ᱟᱨ ᱠᱚᱭᱢᱟ ᱢᱟᱨᱟᱝ ᱦᱟᱹᱱᱤ ᱛᱟᱞᱟ ᱨᱮ ᱪᱮᱫ ᱞᱮᱠᱟ ᱦᱟᱹᱴᱤᱧᱚᱜ-ᱟ᱾ ᱥᱤᱢᱟᱹ ᱵᱮᱜᱚᱨ ᱟᱨ ᱢᱩᱪᱟᱹᱫ ᱵᱟᱹᱱᱩᱜ ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤᱴᱟᱠᱟᱠᱷᱮᱞ ᱯᱩᱸᱡᱤᱠᱷᱮᱞ ᱞᱟᱹᱜᱤᱫ ᱟᱞᱟᱜ ᱫᱚᱦᱚ ᱟᱠᱟᱱ ᱡᱚᱛᱚ ᱴᱟᱠᱟ᱾ ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤ ᱥᱟᱢᱵᱷᱟᱞ ᱫᱚ ᱢᱮᱱᱮᱛ ᱢᱤᱫ ᱵᱟᱡᱤ ᱟᱨ ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱨᱮ ᱚᱱᱟ ᱠᱷᱚᱱ ᱛᱤᱱᱟᱹᱜ ᱡᱚᱠᱷᱤᱢ ᱨᱮ ᱫᱚᱦᱚᱭ ᱚᱱᱟ ᱴᱷᱟᱹᱣᱠᱟᱹ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱥᱟᱶ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱫᱚ “ᱠᱟᱹᱢᱤᱭᱟ”, ᱱᱚᱶᱟ ᱢᱟᱱᱮ ᱨᱮ ᱡᱮ ᱡᱚᱠᱷᱚᱱ ᱦᱚᱸ ᱵᱟᱝ ᱨᱟᱯᱩᱫᱚᱜ-ᱟ, ᱟᱨ ᱚᱱᱟ ᱥᱟᱶ ᱵᱤᱱᱟᱹ ᱥᱤᱢᱟᱹ ᱴᱟᱠᱟ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣᱟ, ᱡᱟᱦᱟᱸ ᱨᱮᱭᱟᱜ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤ ᱫᱚ ᱚᱱᱟ ᱵᱤᱱᱟᱹ ᱥᱤᱢᱟᱹ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ 2.70%᱾ ᱢᱟᱨᱴᱤᱝᱜᱮᱞᱥᱴᱨᱮᱴᱮᱡᱤᱢᱟᱨᱴᱤᱝᱜᱮᱞᱡᱚᱛᱚ ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱵᱟᱡᱤ ᱰᱚᱵᱚᱞ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ᱾ ᱱᱚᱶᱟ ᱪᱮᱫ ᱮ ᱠᱟᱹᱢᱤᱭᱟ ᱟᱨ ᱪᱮᱫ ᱵᱟᱭ ᱠᱟᱹᱢᱤᱭᱟ, ᱚᱱᱟ ᱥᱴᱨᱮᱴᱮᱡᱤ ᱦᱟᱹᱴᱤᱧ ᱨᱮ ᱧᱮᱞ ᱯᱮ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱵᱟᱝ ᱠᱟᱹᱢᱤ ᱨᱮᱭᱟᱜ ᱠᱟᱨᱚᱬ ᱫᱚ ᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹᱴᱮᱵᱩᱞᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹᱦᱩᱠᱩᱢ ᱢᱮᱱᱟᱜ ᱠᱚᱢ ᱥᱮ ᱠᱚᱢ ᱟᱨ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱵᱟᱡᱤ᱾ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱥᱤᱢᱟᱹ ᱜᱮ ᱰᱚᱵᱚᱞ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱵᱚᱸᱫᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱫᱚ ᱵᱟᱝ᱾ ᱦᱤᱥᱟᱹᱵ ᱜᱮ ᱠᱟᱱᱟ᱾

$10 ᱮᱦᱚᱵ ᱛᱮ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ, 100 ᱜᱷᱩᱨᱟᱹᱣ: ᱥᱤᱢᱟᱹ ᱞᱮᱠᱷᱟ ᱛᱮ ᱟᱣᱥᱚᱛ ᱨᱮᱡᱟᱞᱴ (ᱥᱤᱢᱩᱞᱮᱥᱚᱱ)$
$500 ᱥᱤᱢᱟᱹ
≈ −$75
$1,000 ᱥᱤᱢᱟᱹ
≈ −$75
$2,000 ᱥᱤᱢᱟᱹ
≈ −$72
$5,000 ᱥᱤᱢᱟᱹ
≈ −$70
ᱢᱤᱫ ᱞᱮᱠᱟᱱ $10, ᱡᱟᱦᱟᱸ ᱥᱤᱢᱟᱹ
$27
$
04080
ᱵᱟᱨ ᱠᱚ ᱦᱟᱹᱱᱤ ᱨᱮᱭᱟᱜ ᱢᱟᱯ ᱩᱫᱩᱜᱟ᱾ ᱥᱤᱢᱟᱹ ᱪᱮᱛᱟᱱ ᱛᱮ ᱟᱹᱜᱩ ᱛᱮ ᱟᱣᱥᱚᱛ ᱫᱚ ᱯᱷᱟᱞᱟᱹᱣ ᱜᱮ ᱵᱟᱝ ᱥᱮᱞᱮᱫᱚᱜ-ᱟ; ᱪᱮᱛᱟᱱ ᱥᱤᱢᱟᱹ ᱡᱟᱦᱟᱸ ᱦᱚᱸ ᱦᱩᱭᱩᱜ, ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱫᱚ 100 ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱠᱚᱢ ᱵᱮᱥᱤ $2,800 ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣᱟ᱾

ᱵᱟᱨ ᱥᱟᱹᱨᱤ ᱴᱮᱵᱩᱞ

ᱛᱷᱚᱨᱯ (Thorp) ᱨᱮᱭᱟᱜ ᱫᱟᱹᱭᱠᱟᱹ ᱫᱚ $1,000 ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱥᱟᱶ ᱢᱤᱫ $1 ᱴᱮᱵᱩᱞ ᱠᱟᱱᱟ, ᱡᱟᱦᱟᱸ ᱨᱮ ᱢᱤᱫ ᱠᱷᱮᱞᱚᱸᱰ ᱫᱚ ᱢᱤᱫ ᱢᱤᱫ ᱰᱚᱞᱟᱨ ᱛᱮ ᱧᱤᱫᱟᱹ ᱫᱟᱠᱟ ᱞᱟᱹᱜᱤᱫ $30 ᱡᱤᱛᱟᱹᱣ ᱞᱟᱹᱜᱤᱫ ᱮ ᱟᱥᱟ ᱮᱫᱟ᱾ ᱵᱟᱡᱤ ᱴᱟᱠᱟ 1, 2, 4 … 512 ᱫᱚ ᱜᱮᱞ ᱵᱟᱡᱤ ᱦᱩᱠᱩᱢ ᱮᱢᱟ, ᱚᱱᱟᱛᱮ ᱞᱟᱜᱟᱛᱟᱨ ᱜᱮᱞ ᱦᱟᱹᱱᱤ ᱛᱮ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱨᱟᱯᱩᱫᱚᱜ-ᱟ: ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 785 ᱪᱟᱠᱨᱟ ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ (ᱟᱢᱮᱨᱤᱠᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 613 ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ), $1,023 ᱠᱷᱚᱨᱚᱪ ᱥᱟᱶ᱾ ᱯᱮ ᱜᱮᱞ ᱪᱟᱠᱨᱟ ᱡᱚᱛᱚ ᱥᱟᱯᱷᱞ ᱦᱩᱭᱩᱜ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱫᱚ 96.2%᱾ ᱵᱟᱠᱤ 3.8% ᱨᱮ ᱠᱚᱢ ᱵᱮᱥᱤ $1,000 ᱦᱟᱨᱟᱣᱚᱜ-ᱟ, ᱟᱨ ᱧᱤᱫᱟᱹ ᱨᱮᱭᱟᱜ ᱟᱣᱥᱚᱛ ᱨᱮᱡᱟᱞᱴ ᱫᱚ −$9.17, ᱡᱟᱦᱟᱸ ᱫᱚ ᱯᱮ ᱜᱮᱞ ᱪᱟᱠᱨᱟ ᱨᱮ ᱟᱣᱥᱚᱛ ᱛᱮ ᱞᱟᱜᱟᱣ $339 ᱨᱮᱭᱟᱜ ᱴᱷᱤᱠ 1/37᱾ $2,000 ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱥᱟᱶ $5 ᱴᱮᱵᱩᱞ ᱦᱚᱸ ᱵᱷᱮᱜᱟᱨ ᱫᱚ ᱵᱟᱝ: $1,280 ᱦᱟᱹᱵᱤᱡ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ ᱟᱨᱮ ᱵᱟᱡᱤ ᱦᱩᱠᱩᱢ ᱮᱢᱟ, ᱟᱨ ᱨᱟᱯᱩᱫ ᱫᱚ ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 403 ᱪᱟᱠᱨᱟ ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ (ᱟᱢᱮᱨᱤᱠᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 323 ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ) ᱦᱩᱭᱩᱜ-ᱟ ᱟᱨ $2,555 ᱠᱷᱚᱨᱚᱪ ᱮᱢᱟ᱾

ᱵᱟᱨ ᱫᱷᱟᱣ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ ᱴᱮᱵᱩᱞ ᱨᱮᱭᱟᱜ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱥᱮᱫ ᱪᱮᱛᱟᱱ ᱛᱮ ᱞᱟᱜᱟᱣ ᱠᱟᱱᱟ, ᱟᱨ ᱫᱟᱨᱟᱭ ᱠᱩᱴᱩᱢ ᱨᱮ ᱧᱤᱫᱟᱹ ᱫᱟᱠᱟ ᱟᱹᱛᱩᱨ ᱠᱟᱱᱟ᱾

ᱢᱤᱫ ᱫᱷᱚᱱᱤ ᱠᱷᱮᱞᱚᱸᱰ

ᱠᱚᱢ ᱥᱤᱢᱟᱹ ᱵᱚᱫᱚᱞ ᱛᱮ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱫᱚ ᱢᱤᱫ ᱢᱟᱨᱟᱝ ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤ ᱮᱢ ᱯᱮ: 1,000 ᱭᱩᱱᱤᱴ, ᱢᱤᱫ ᱭᱩᱱᱤᱴ ᱮᱦᱚᱵ ᱟᱨ ᱟᱨᱟᱜ ᱨᱮ 1,000 ᱜᱷᱩᱨᱟᱹᱣ, ᱫᱟᱨᱟᱭ ᱵᱟᱨ ᱫᱷᱟᱣ ᱫᱚ ᱵᱟᱠᱤ ᱠᱷᱚᱱ ᱵᱮᱥᱤ ᱦᱩᱭᱩᱜ ᱠᱷᱟᱱ ᱫᱚᱦᱲᱟ ᱢᱤᱫ ᱭᱩᱱᱤᱴ ᱠᱷᱚᱱ ᱮᱦᱚᱵ ᱠᱟᱛᱮ᱾ ᱟᱞᱮ ᱢᱤᱫ ᱞᱟᱠᱷ ᱱᱚᱶᱟ ᱞᱮᱠᱟᱱ ᱧᱤᱫᱟᱹ ᱥᱤᱢᱩᱞᱮᱥᱚᱱ ᱞᱮ ᱠᱟᱹᱢᱤ ᱠᱮᱫᱟ᱾ ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 52% ᱞᱟᱵᱷ ᱨᱮ ᱢᱩᱪᱟᱹᱫᱚᱜ-ᱟ, 21% ᱡᱚᱛᱚ ᱦᱟᱨᱟᱣᱟ, ᱟᱨ ᱟᱣᱥᱚᱛ ᱨᱮᱡᱟᱞᱴ ᱫᱚ −124 ᱭᱩᱱᱤᱴ᱾ ᱡᱤᱨᱚᱪᱟᱠᱟᱡᱤᱨᱚᱢᱤᱫᱜᱮ ᱦᱟᱹᱨᱭᱟᱹᱲ ᱠᱷᱟᱱᱟ ᱡᱟᱦᱟᱸ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ, ᱰᱚᱡᱮᱱ ᱟᱨᱵᱟᱝ ᱠᱚᱞᱚᱢ ᱵᱟᱡᱤ ᱵᱟᱭ ᱠᱚᱵᱷᱟᱨᱟ; ᱠᱮᱥᱤᱱᱳ ᱵᱟᱲᱛᱤ ᱨᱮᱭᱟᱜ ᱢᱩᱞ᱾ ᱮᱴᱟᱜ ᱧᱩᱛᱩᱢ: ᱥᱤᱝᱜᱚᱞ ᱡᱤᱨᱚ, 0᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱵᱮᱜᱚᱨ ᱠᱷᱮᱞ ᱨᱮ ᱦᱚᱸ, ᱡᱟᱦᱟᱸ ᱨᱮ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱫᱚ ᱱᱤᱭᱟᱭ, ᱠᱷᱟᱹᱞᱤ ᱠᱚᱢ ᱵᱮᱥᱤ 60% ᱜᱮ ᱞᱟᱵᱷ ᱨᱮ ᱢᱩᱪᱟᱹᱫᱚᱜ-ᱟ᱾ ᱥᱟᱫᱷᱟᱨᱚᱬ ᱧᱤᱫᱟᱹ ᱫᱚ ᱦᱩᱰᱤᱧ ᱡᱤᱛ ᱠᱟᱱᱟ; ᱟᱣᱥᱚᱛ ᱨᱮᱭᱟᱜ ᱫᱟᱢ ᱫᱚ ᱡᱤᱨᱚ ᱨᱮ ᱢᱩᱪᱟᱹᱫ ᱧᱤᱫᱟᱹ ᱠᱚ ᱮᱢᱟ᱾

ᱪᱟᱠᱟᱞᱟᱵᱷ ᱨᱮ ᱢᱩᱪᱟᱹᱫ1,000 ᱭᱩᱱᱤᱴ ᱦᱟᱨᱟᱣᱟᱣᱥᱚᱛ ᱨᱮᱡᱟᱞᱴ
ᱡᱤᱨᱚ ᱵᱮᱜᱚᱨ (ᱱᱤᱭᱟᱭ ᱠᱷᱮᱞ)60%15%0
ᱭᱩᱨᱚᱯᱤᱭᱟᱱ52%21%−124 ᱭᱩᱱᱤᱴ
ᱟᱢᱮᱨᱤᱠᱟᱱ45%30%−250 ᱭᱩᱱᱤᱴ
ᱟᱞᱮᱭᱟᱜ ᱥᱤᱢᱩᱞᱮᱥᱚᱱ, ᱡᱚᱛᱚ ᱪᱟᱠᱟ ᱞᱟᱹᱜᱤᱫ ᱢᱤᱫ ᱞᱟᱠᱷ ᱧᱤᱫᱟᱹ᱾ ᱟᱣᱥᱚᱛ ᱫᱚ 1,000 ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱞᱟᱜᱟᱣ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ 1/37 ᱟᱨ 1/19: ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱪᱟᱠᱟ ᱨᱮ ᱠᱚᱢ ᱵᱮᱥᱤ 4,600 ᱭᱩᱱᱤᱴ ᱟᱨ ᱟᱢᱮᱨᱤᱠᱟᱱ ᱨᱮ 4,750᱾

ᱥᱮᱪᱮᱫ

ᱠᱮᱥᱤᱱᱳ ᱠᱚ ᱫᱚ ᱟᱡᱟᱜ ᱣᱮᱨᱤᱭᱮᱱᱥᱜᱚᱱᱤᱛᱣᱮᱨᱤᱭᱮᱱᱥᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱜᱷᱮᱨᱟᱣ ᱨᱮᱡᱟᱞᱴ ᱛᱤᱱᱟᱹᱜ ᱯᱟᱥᱱᱟᱣᱚᱜ-ᱟ ᱚᱱᱟ ᱨᱮᱭᱟᱜ ᱥᱟᱯᱲᱟᱣ᱾ ᱥᱴᱨᱮᱴ ᱚᱯ ᱞᱟᱹᱜᱤᱫ ᱵᱟᱹᱲᱛᱤ, ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ ᱞᱟᱹᱜᱤᱫ ᱠᱚᱢ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱛᱷᱤᱨ ᱞᱟᱹᱜᱤᱫ ᱥᱤᱢᱟᱹ ᱠᱚ ᱫᱚᱦᱚᱭᱟ, ᱥᱤᱥᱴᱮᱢ ᱦᱟᱨᱟᱣ ᱞᱟᱹᱜᱤᱫ ᱫᱚ ᱵᱟᱝ᱾ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱦᱟᱨᱟᱣ ᱫᱚ ᱢᱤᱫ ᱫᱷᱟᱨᱮ ᱨᱮᱡᱟᱞᱴ ᱠᱟᱱᱟ ᱡᱟᱦᱟᱸ ᱛᱮ ᱩᱱᱠᱩ ᱠᱩᱥᱤ, ᱟᱨ ᱦᱤᱥᱟᱹᱵ ᱫᱚ ᱚᱱᱟ ᱡᱮᱞᱮᱠᱟ ᱦᱚᱸ ᱦᱟᱨᱟᱣ ᱜᱮᱭᱟ᱾ ᱡᱟᱦᱟᱸ ᱠᱷᱮᱞᱚᱸᱰ ᱵᱷᱟᱵᱱᱟᱭ ᱡᱮ ᱥᱤᱢᱟᱹ ᱜᱮ ᱥᱚᱢᱥᱭᱟ ᱠᱟᱱᱟ, ᱩᱱᱤ ᱫᱚ ᱫᱷᱤᱨᱤ ᱠᱟᱲᱟᱹᱠ ᱮ ᱧᱟᱢ ᱟᱠᱟᱫᱟ ᱟᱨ ᱚᱱᱟ ᱫᱚ ᱞᱟᱛᱟᱨ ᱥᱮᱫ ᱪᱟᱞᱟᱜ ᱦᱚᱨ ᱞᱮᱠᱟ ᱮ ᱵᱷᱩᱞ ᱵᱩᱡᱷᱟᱹᱣ ᱟᱠᱟᱫᱟ᱾

ᱯᱷᱚᱨᱢᱩᱞᱟ

p(ᱨᱟᱯᱩᱫ) = (19/37)ᵏ, k = ᱥᱤᱢᱟᱹ ᱦᱩᱠᱩᱢ ᱮᱢ ᱦᱟᱹᱱᱤ
ᱨᱟᱯᱩᱫ ᱨᱮ ᱦᱟᱹᱱᱤ = ᱮᱦᱚᱵ · (2ᵏ − 1)
E[ᱢᱤᱫ ᱪᱟᱠᱨᱟ ᱨᱮ ᱦᱟᱹᱱᱤ] = E[ᱪᱟᱠᱨᱟ ᱨᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ] / 37

ᱠᱩᱠᱞᱤ ᱠᱚ

ᱢᱟᱨᱟᱝ ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤ ᱫᱚ ᱱᱚᱶᱟ ᱴᱷᱤᱠᱟᱹᱣᱟ ᱥᱮ?

ᱚᱱᱟ ᱫᱚ ᱟᱯᱮ ᱪᱮᱛᱟᱱ ᱥᱤᱢᱟᱹ ᱦᱟᱹᱵᱤᱡ ᱵᱟᱧᱪᱟᱣ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ ᱯᱮ, ᱡᱟᱦᱟᱸ ᱨᱮ ᱨᱟᱯᱩᱫ ᱫᱚ ᱠᱚᱢ ᱟᱨ ᱢᱟᱨᱟᱝ ᱦᱩᱭᱩᱜ-ᱟ᱾ ᱢᱤᱫ ᱰᱚᱞᱟᱨ ᱨᱮ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤ ᱫᱚ ᱵᱟᱝ ᱵᱚᱫᱚᱞᱚᱜ-ᱟ᱾

ᱡᱚᱛᱚ ᱵᱟᱨ ᱫᱷᱟᱣ ᱨᱮ ᱢᱤᱫ ᱭᱩᱱᱤᱴ ᱥᱮᱞᱮᱫ “ᱜᱽᱨᱮᱱᱰ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ” ᱫᱚ?

ᱵᱟᱡᱤ ᱴᱟᱠᱟ 10, 30, 70, 150, 310: ᱢᱚᱬᱮ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ $500 ᱥᱤᱢᱟᱹ ᱥᱮᱴᱮᱨᱚᱜ-ᱟ, ᱟᱨ ᱨᱟᱯᱩᱫ ᱨᱮᱭᱟᱜ ᱠᱷᱚᱨᱚᱪ $570᱾ ᱚᱱᱟ ᱜᱮ 2.70%᱾

ᱠᱮᱥᱤᱱᱳ ᱠᱚ ᱨᱮ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱥᱤᱢᱟᱹ ᱪᱮᱫᱟᱜ ᱢᱮᱱᱟᱜ-ᱟ?

ᱡᱮᱢᱚᱱ ᱢᱤᱫ ᱠᱷᱮᱞᱚᱸᱰ ᱫᱚ ᱠᱮᱥᱤᱱᱳ ᱨᱮᱭᱟᱜ ᱡᱚᱢᱟ ᱴᱟᱠᱟ ᱟᱨᱟᱢ ᱛᱮ ᱮᱢ ᱫᱟᱲᱮᱭᱟᱜ ᱠᱷᱚᱱ ᱵᱮᱥᱤ ᱵᱟᱡᱤ ᱟᱞᱚᱭ ᱞᱟᱜᱟᱣ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ᱾ ᱣᱮᱨᱤᱭᱮᱱᱥ ᱥᱟᱢᱵᱷᱟᱞ᱾

ᱜᱚᱱᱤᱛ ᱥᱟᱶ ᱨᱟᱹᱥᱠᱟᱹ

ᱱᱚᱶᱟ ᱠᱚ ᱯᱩᱭᱞᱩ ᱟᱯᱮ ᱛᱮ ᱠᱩᱨᱩᱢᱩᱴᱩ ᱯᱮ᱾ ᱢᱤᱫ ᱢᱤᱫ ᱥᱚᱢᱟᱫᱷᱟᱱ ᱫᱚ ᱩᱠᱩ ᱢᱮᱱᱟᱜ-ᱟ; ᱛᱮᱭᱟᱨ ᱦᱩᱭ ᱠᱟᱛᱮ ᱡᱷᱤᱡᱽ ᱯᱮ᱾

  1. ᱢᱤᱫ ᱢᱟᱨᱴᱤᱝᱜᱮᱞᱥᱴᱨᱮᱴᱮᱡᱤᱢᱟᱨᱴᱤᱝᱜᱮᱞᱡᱚᱛᱚ ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱵᱟᱡᱤ ᱰᱚᱵᱚᱞ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ᱾ ᱱᱚᱶᱟ ᱪᱮᱫ ᱮ ᱠᱟᱹᱢᱤᱭᱟ ᱟᱨ ᱪᱮᱫ ᱵᱟᱭ ᱠᱟᱹᱢᱤᱭᱟ, ᱚᱱᱟ ᱥᱴᱨᱮᱴᱮᱡᱤ ᱦᱟᱹᱴᱤᱧ ᱨᱮ ᱧᱮᱞ ᱯᱮ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱟᱨᱟᱜᱵᱟᱡᱤᱟᱨᱟᱜ18 ᱟᱨᱟᱜ ᱮᱞ ᱨᱮ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ᱾ ᱯᱷᱨᱮᱧᱪ ᱨᱮ: ᱨᱩᱡᱽ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱮ $100 ᱠᱷᱚᱱ ᱮᱦᱚᱵᱚᱜ-ᱟ ᱟᱨ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ n ᱵᱟᱡᱤ ᱮ ᱟᱹᱜᱩᱭᱟ: ᱞᱟᱜᱟᱛᱟᱨ n ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱚᱱᱟ ᱮ ᱵᱟᱹᱜᱤᱭᱟ ᱟᱨ ᱫᱚᱦᱲᱟ ᱮ ᱮᱦᱚᱵᱟ᱾ ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱨᱮᱭᱟᱜ ᱟᱣᱥᱚᱛ ᱨᱮᱡᱟᱞᱴ n ᱨᱮ ᱯᱷᱚᱨᱢᱩᱞᱟ ᱞᱮᱠᱟᱛᱮ ᱚᱞ ᱯᱮ, ᱟᱨ ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 5, 6 ᱟᱨ 7 ᱵᱟᱡᱤ ᱞᱟᱹᱜᱤᱫ ᱦᱤᱥᱟᱹᱵ ᱯᱮ᱾

    ᱥᱚᱢᱟᱫᱷᱟᱱ ᱩᱫᱩᱜ

    ᱛᱮᱞᱟ ᱟᱣᱥᱚᱛ ᱫᱚ 100 × (1 − (38/37)ⁿ): 5 ᱵᱟᱡᱤ ᱞᱟᱹᱜᱤᱫ −$14.26, 6 ᱞᱟᱹᱜᱤᱫ −$17.35 ᱟᱨ 7 ᱞᱟᱹᱜᱤᱫ −$20.52᱾ ᱵᱮᱥᱤ ᱵᱟᱡᱤ ᱟᱹᱜᱩ ᱛᱮ ᱟᱣᱥᱚᱛ ᱪᱚᱠᱨᱚ ᱫᱚ ᱵᱟᱹᱲᱤᱡ ᱦᱩᱭᱩᱜ-ᱟ, ᱵᱮᱥ ᱫᱚ ᱵᱟᱝ᱾

    ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱫᱚ $100 ᱡᱤᱛᱟᱹᱣᱚᱜ-ᱟ ᱡᱩᱫᱤ ᱡᱚᱛᱚ n ᱵᱟᱡᱤ ᱵᱟᱝ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ, ᱚᱱᱟ ᱫᱚ qⁿ ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱛᱮ ᱦᱩᱭᱩᱜ-ᱟ, ᱡᱟᱦᱟᱸ ᱨᱮ q = 19/37᱾ ᱡᱚᱠᱷᱚᱱ ᱡᱚᱛᱚ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ, ᱵᱟᱡᱤ ᱴᱟᱠᱟ 100, 200 … 100 × 2ⁿ⁻¹ ᱡᱚᱲᱟᱣ ᱠᱟᱛᱮ 100 × (2ⁿ − 1) ᱦᱩᱭᱩᱜ-ᱟ᱾ ᱚᱱᱟᱛᱮ ᱟᱣᱥᱚᱛ ᱫᱚ 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ)᱾

    ᱱᱚᱰᱮ 2q = 38/37, 1 ᱠᱷᱚᱱ ᱛᱤᱱᱟᱹᱜ ᱜᱟᱱ ᱵᱮᱥᱤ, ᱚᱱᱟᱛᱮ (2q)ⁿ ᱫᱚ n ᱥᱟᱶ ᱵᱟᱲᱦᱟᱣᱚᱜ-ᱟ ᱟᱨ ᱟᱣᱥᱚᱛ ᱞᱟᱛᱟᱨ ᱥᱮᱫ ᱪᱟᱞᱟᱜ-ᱟ: n = 5 ᱫᱚ 1.1426 ᱟᱨ −$14.26 ᱮᱢᱟ; n = 6 ᱫᱚ 1.1735 ᱟᱨ −$17.35; n = 7 ᱫᱚ 1.2052 ᱟᱨ −$20.52᱾ ᱡᱮᱞᱮᱧ ᱥᱤᱲᱤ ᱫᱚ ᱠᱚᱢ ᱫᱷᱟᱣ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ, 3.6%, 1.8% ᱟᱨ 0.9% ᱪᱚᱠᱨᱚ ᱨᱮ, ᱢᱮᱱᱠᱷᱟᱱ ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱵᱟᱨ ᱫᱷᱟᱣ ᱢᱟᱨᱟᱝ, ᱟᱨ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮ ᱵᱮᱥᱤ ᱴᱟᱠᱟ ᱴᱮᱵᱩᱞ ᱪᱮᱛᱟᱱ ᱛᱮ ᱪᱟᱞᱟᱜ-ᱟ, ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ 2.70% ᱨᱮ᱾

    ᱟᱢᱮᱨᱤᱠᱟᱱ ᱪᱟᱠᱟ ᱨᱮ 2q = 40/38 ᱟᱨ ᱮᱞ ᱠᱚ ᱫᱚ −$29.24, −$36.04 ᱟᱨ −$43.20 (ᱴᱮᱞᱟᱨ, Taylor, 2021)᱾ ᱡᱤᱨᱚᱪᱟᱠᱟᱡᱤᱨᱚᱢᱤᱫᱜᱮ ᱦᱟᱹᱨᱭᱟᱹᱲ ᱠᱷᱟᱱᱟ ᱡᱟᱦᱟᱸ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ, ᱰᱚᱡᱮᱱ ᱟᱨᱵᱟᱝ ᱠᱚᱞᱚᱢ ᱵᱟᱡᱤ ᱵᱟᱭ ᱠᱚᱵᱷᱟᱨᱟ; ᱠᱮᱥᱤᱱᱳ ᱵᱟᱲᱛᱤ ᱨᱮᱭᱟᱜ ᱢᱩᱞ᱾ ᱮᱴᱟᱜ ᱧᱩᱛᱩᱢ: ᱥᱤᱝᱜᱚᱞ ᱡᱤᱨᱚ, 0᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱵᱟᱹᱱᱩᱜ ᱠᱷᱮᱞ ᱨᱮ 2q = 1, ᱟᱨ ᱥᱤᱲᱤ ᱡᱮᱛᱤᱠ ᱦᱚᱸ ᱡᱮᱞᱮᱧ ᱛᱟᱦᱮᱸᱱ ᱠᱷᱟᱱ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮᱭᱟᱜ ᱟᱣᱥᱚᱛ ᱴᱷᱤᱠ ᱡᱤᱨᱚ ᱜᱮ᱾

  2. “ᱜᱽᱨᱮᱱᱰ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ” ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱵᱟᱨ ᱫᱷᱟᱣ ᱮ ᱵᱮᱱᱟᱣᱟ ᱟᱨ ᱢᱤᱫ ᱭᱩᱱᱤᱴᱴᱟᱠᱟᱭᱩᱱᱤᱴᱢᱤᱫ ᱥᱤᱥᱴᱮᱢ ᱟᱨᱵᱟᱝ ᱠᱷᱮᱞᱚᱸᱰ ᱵᱮᱵᱷᱟᱨ ᱵᱮᱥ ᱵᱟᱡᱤ ᱴᱟᱠᱟ᱾ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱠᱚ ᱭᱩᱱᱤᱴ ᱛᱮ ᱵᱩᱡᱷᱟᱹᱣᱚᱜ-ᱟ ᱡᱮᱢᱚᱱ ᱡᱟᱦᱟᱸᱱ ᱴᱮᱵᱩᱞ ᱨᱮ ᱞᱟᱹᱜᱩ ᱦᱩᱭᱩᱜ-ᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱮ ᱥᱮᱞᱮᱫᱟ, ᱚᱱᱟᱛᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ 1, 3, 7, 15, 31 … ᱞᱮᱠᱟ ᱪᱟᱞᱟᱜ-ᱟ᱾ ᱯᱟᱦᱤᱞ ᱡᱤᱛ 1, 2, 3 ᱥᱮ k ᱱᱚᱢᱵᱚᱨ ᱵᱟᱡᱤ ᱨᱮ ᱦᱤᱡᱩᱜ ᱠᱷᱟᱱ ᱟᱯᱮ ᱛᱤᱱᱟᱹᱜ ᱞᱟᱵᱷ ᱨᱮ ᱢᱮᱱᱟᱜ-ᱟ ᱯᱮ? $10 ᱭᱩᱱᱤᱴ ᱟᱨ $500 ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱨᱮ, ᱪᱚᱠᱨᱚ ᱛᱤᱱᱟᱹᱜ ᱫᱷᱟᱣ ᱨᱟᱹᱯᱩᱫᱚᱜ-ᱟ, ᱟᱨ ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱨᱮᱭᱟᱜ ᱟᱣᱥᱚᱛ ᱠᱷᱚᱨᱚᱪ ᱫᱚ ᱪᱮᱫ?

    ᱥᱚᱢᱟᱫᱷᱟᱱ ᱩᱫᱩᱜ

    ᱛᱮᱞᱟ k ᱱᱚᱢᱵᱚᱨ ᱵᱟᱡᱤ ᱨᱮ ᱡᱤᱛ ᱦᱩᱭ ᱠᱷᱟᱱ ᱟᱯᱮ k ᱭᱩᱱᱤᱴ ᱞᱟᱵᱷ ᱨᱮ ᱛᱟᱦᱮᱸᱭᱟ ᱯᱮ: +1, +2, +3 ᱟᱨ ᱚᱱᱟ ᱞᱮᱠᱟ ᱜᱮ᱾ $10 ᱟᱨ $500 ᱨᱮ ᱪᱚᱠᱨᱚ ᱫᱚ ᱢᱚᱬᱮ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱨᱟᱹᱯᱩᱫᱚᱜ-ᱟ, 3.57% ᱪᱚᱠᱨᱚ ᱨᱮ, −$570 ᱞᱟᱹᱜᱤᱫ᱾ ᱟᱣᱥᱚᱛ ᱛᱮ ᱢᱤᱫ ᱪᱚᱠᱨᱚ $85.73 ᱵᱟᱡᱤ ᱠᱷᱚᱱ $2.32 ᱦᱟᱨᱟᱣᱚᱜ-ᱟ: 2.70%᱾

    k ᱱᱚᱢᱵᱚᱨ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ 2ᵏ − 1 ᱭᱩᱱᱤᱴ: 1, 3, 7, 15, 31᱾ ᱯᱟᱦᱤᱞ k − 1 ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱡᱚᱲᱟᱣ ᱠᱟᱛᱮ 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1᱾ k ᱱᱚᱢᱵᱚᱨ ᱵᱟᱡᱤ ᱨᱮ ᱡᱤᱛ 2ᵏ − 1 ᱮᱢᱟ, ᱚᱱᱟᱛᱮ ᱪᱚᱠᱨᱚ ᱫᱚ (2ᵏ − 1) − (2ᵏ − k − 1) = k ᱭᱩᱱᱤᱴ ᱞᱟᱵᱷ ᱨᱮ ᱢᱩᱪᱟᱹᱫᱚᱜ-ᱟ: ᱡᱤᱛ ᱞᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱞᱟᱹᱜᱤᱫ ᱢᱤᱫ ᱵᱮᱥᱤ ᱭᱩᱱᱤᱴ᱾

    $10 ᱭᱩᱱᱤᱴ ᱨᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ $10, $30, $70, $150 ᱟᱨ $310; ᱫᱟᱨᱟᱭ, $630, ᱫᱚ $500 ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱠᱷᱚᱱ ᱵᱮᱥᱤ᱾ ᱞᱟᱜᱟᱛᱟᱨ ᱢᱚᱬᱮ ᱦᱟᱹᱱᱤ, (19/37)⁵ = 3.57%, ᱨᱮᱭᱟᱜ ᱠᱷᱚᱨᱚᱪ 10 + 30 + 70 + 150 + 310 = $570᱾

    ᱟᱣᱥᱚᱛ: ᱡᱤᱛ ᱫᱚ k = 1 ᱠᱷᱚᱱ 5 ᱦᱟᱹᱵᱤᱡ (19/37)ᵏ⁻¹ × 18/37 ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱛᱮ $10 × k ᱮ ᱟᱹᱜᱩᱭᱟ, ᱟᱨ ᱨᱟᱹᱯᱩᱫ ᱫᱚ 3.57% ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱛᱮ $570 ᱮ ᱤᱫᱤᱭᱟ; ᱥᱟᱶᱛᱮ −$2.32᱾ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱟᱣᱥᱚᱛ $85.73, ᱟᱨ 2.32/85.73 = 2.70%᱾ ᱚᱱᱟ ᱜᱮ ᱴᱮᱵᱩᱞ ᱨᱮ ᱥᱟᱫᱷᱟᱨᱚᱬ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱫᱚ ᱛᱩᱨᱩᱭ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱨᱟᱹᱯᱩᱫᱚᱜ-ᱟ, 1.83% ᱪᱚᱠᱨᱚ ᱨᱮ, −$630 ᱞᱟᱹᱜᱤᱫ, ᱟᱨ $64.20 ᱵᱟᱡᱤ ᱠᱷᱚᱱ $1.74 ᱦᱟᱨᱟᱣᱚᱜ-ᱟ: ᱟᱨᱦᱚᱸ 2.70%᱾

  3. ᱢᱤᱫ ᱠᱷᱮᱞᱚᱸᱰ ᱟᱨᱟᱜ ᱨᱮ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱨᱮᱭᱟᱜ ᱯᱞᱟᱱ ᱮ ᱵᱮᱱᱟᱣᱟ, $1 ᱯᱟᱦᱤᱞ ᱵᱟᱡᱤ ᱥᱟᱶ, ᱡᱟᱦᱟᱸ ᱴᱮᱵᱩᱞ ᱨᱮᱭᱟᱜ ᱥᱤᱢᱟᱹ $1,000 ᱚᱱᱰᱮ, ᱚᱱᱟᱛᱮ ᱜᱮᱞ ᱵᱟᱡᱤ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ: $1, $2, $4 ᱟᱨ ᱚᱱᱟ ᱞᱮᱠᱟ $512 ᱦᱟᱹᱵᱤᱡ᱾ ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱛᱤᱱᱟᱹᱜ ᱫᱷᱟᱣ ᱡᱚᱛᱚ ᱜᱮᱞ ᱦᱟᱨᱟᱣᱟ, ᱚᱱᱟ ᱞᱟᱹᱜᱤᱫ ᱟᱣᱥᱚᱛ ᱛᱮ ᱛᱤᱱᱟᱹᱜ ᱚᱠᱛᱚ ᱞᱟᱹᱠᱛᱤᱭᱟ, ᱟᱨ 200 ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ ᱢᱤᱫ ᱧᱤᱫᱟᱹ ᱛᱤᱱᱟᱹᱜ ᱫᱷᱟᱣ ᱚᱱᱟ ᱞᱮᱠᱟᱛᱮ ᱢᱩᱪᱟᱹᱫᱚᱜ-ᱟ?

    ᱥᱚᱢᱟᱫᱷᱟᱱ ᱩᱫᱩᱜ

    ᱛᱮᱞᱟ ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱡᱟᱦᱟᱱ 784 ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ ᱡᱚᱛᱚ ᱜᱮᱞ ᱦᱟᱨᱟᱣᱟ, ᱚᱱᱟ ᱨᱮᱭᱟᱜ ᱠᱷᱚᱨᱚᱪ $1,023᱾ ᱚᱱᱟ ᱫᱚ ᱟᱣᱥᱚᱛ ᱛᱮ ᱡᱟᱦᱟᱱ 1,600 ᱵᱟᱡᱤ, ᱟᱨ 200 ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ ᱧᱤᱫᱟᱹ ᱡᱟᱦᱟᱱ 11% ᱚᱠᱛᱚ ᱵᱷᱤᱛ ᱨᱮ ᱫᱷᱟᱠᱟᱣᱚᱜ-ᱟ᱾ ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱟᱣᱥᱚᱛ ᱛᱮ ᱟᱡᱟᱦᱟᱸ $11.31 ᱵᱟᱡᱤ ᱠᱷᱚᱱ $0.31 ᱦᱟᱨᱟᱣᱚᱜ-ᱟ: 2.70%᱾

    ᱟᱨᱟᱜ ᱨᱮ ᱵᱟᱡᱤ 19/37 ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱛᱮ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ, ᱡᱤᱨᱚ ᱥᱮᱞᱮᱫ ᱠᱟᱛᱮ᱾ ᱞᱟᱜᱟᱛᱟᱨ ᱜᱮᱞ: (19/37)¹⁰ ≈ 0.00128, ᱡᱟᱦᱟᱱ 784 ᱨᱮ 1᱾ ᱵᱟᱡᱤ ᱴᱟᱠᱟ 1 + 2 + 4 + … + 512 ᱡᱚᱲᱟᱣ ᱠᱟᱛᱮ $1,023᱾

    ᱢᱤᱫ ᱪᱚᱠᱨᱚ ᱟᱣᱥᱚᱛ ᱛᱮ 2.05 ᱵᱟᱡᱤ ᱞᱟᱹᱠᱛᱤᱭᱟ, ᱚᱱᱟᱛᱮ 784 ᱪᱚᱠᱨᱚ ᱞᱟᱹᱜᱤᱫ ᱡᱟᱦᱟᱱ 1,600 ᱵᱟᱡᱤ᱾ ᱜᱷᱩᱨᱟᱹᱣ ᱜᱷᱩᱨᱟᱹᱣ ᱞᱮᱠᱷᱟ ᱠᱟᱛᱮ, 200 ᱵᱟᱡᱤ ᱨᱮ ᱡᱟᱦᱟᱸ ᱦᱚᱸ ᱡᱟᱭᱜᱟ ᱨᱮ ᱜᱮᱞ ᱦᱟᱹᱱᱤ ᱨᱮᱭᱟᱜ ᱞᱟᱜᱟᱛᱟᱨ 11.3% ᱚᱠᱛᱚ ᱧᱟᱢᱚᱜ-ᱟ᱾

    ᱵᱤᱯᱚᱫ ᱠᱚ ᱛᱟᱞᱟ ᱨᱮ ᱠᱷᱮᱞᱚᱸᱰ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮ $1 ᱮ ᱡᱤᱛᱟᱹᱣᱟ: ᱵᱟᱹᱲᱤᱡ ᱪᱚᱠᱨᱚ ᱞᱟᱦᱟ ᱨᱮ ᱡᱟᱦᱟᱱ $783, ᱛᱟᱭᱚᱢ −$1,023, ᱜᱚᱴᱟ ᱚᱠᱛᱚ ᱨᱮ ᱡᱟᱦᱟᱱ −$240᱾ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮ: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31᱾ ᱡᱚᱛᱚ ᱪᱚᱠᱨᱚ ᱨᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31᱾ ᱟᱨ −0.31/11.31 = −2.70%᱾

    $500 ᱥᱤᱢᱟᱹ ᱨᱮ $10 ᱯᱟᱦᱤᱞ ᱵᱟᱡᱤ ᱥᱟᱶ ᱛᱩᱞᱟᱹᱣ ᱯᱮ, ᱛᱩᱨᱩᱭ ᱵᱟᱡᱤ: ᱵᱷᱤᱛ ᱫᱚ 55 ᱪᱚᱠᱨᱚ ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ ᱦᱮᱡᱚᱜ-ᱟ, ᱟᱨ 200 ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ ᱧᱤᱫᱟᱹ 84% ᱚᱠᱛᱚ ᱚᱱᱟ ᱥᱟᱶ ᱧᱟᱯᱟᱢᱚᱜ-ᱟ᱾ ᱢᱟᱨᱟᱝ ᱥᱤᱢᱟᱹ ᱫᱚ ᱨᱟᱹᱯᱩᱫ ᱠᱚᱢ ᱟᱨ ᱢᱟᱨᱟᱝ ᱮ ᱵᱮᱱᱟᱣᱟ, ᱥᱚᱥᱛᱟ ᱫᱚ ᱵᱟᱝ᱾

ᱞᱟᱸᱫᱟ ᱞᱟᱹᱜᱤᱫ ᱠᱚᱢᱤᱠ

ᱡᱚᱛᱚ ᱥᱤᱢᱟᱹ ᱨᱮ ᱠᱚᱢ ᱫᱷᱟᱣ ᱟᱨ ᱵᱮᱥᱤ ᱢᱟᱨᱟᱝ᱾ ᱜᱩᱬ ᱫᱚ ᱛᱷᱤᱨ ᱜᱮ᱾

ᱡᱚᱛᱚ ᱥᱤᱢᱟᱹ ᱨᱮ ᱠᱚᱢ ᱫᱷᱟᱣ ᱟᱨ ᱵᱮᱥᱤ ᱢᱟᱨᱟᱝ᱾ ᱜᱩᱬ ᱫᱚ ᱛᱷᱤᱨ ᱜᱮ᱾

ᱯᱷᱮᱰᱟᱛ

  1. Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010᱾ ᱪᱮᱯᱴᱟᱨ 13, ᱨᱩᱞᱮᱴ᱾
  2. Thorp, E. O. “Optimal gambling systems for favorable games.” Review of the International Statistical Institute 37(3), 1969᱾
  3. Thorp, E. O. The Mathematics of Gambling. Gambling Times / Lyle Stuart, 1984᱾
  4. Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013᱾
ᱨᱩᱞᱮᱴ ᱵᱟᱡᱤ ᱥᱴᱨᱮᱴᱮᱡᱤᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱨᱩᱞᱮᱴ ᱥᱴᱨᱮᱴᱮᱡᱤᱨᱩᱞᱮᱴ ᱜᱚᱱᱤᱛᱨᱩᱞᱮᱴ ᱨᱮ ᱞᱟᱜᱟᱛᱟᱨ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱵᱷᱟᱵᱱᱟᱨᱩᱞᱮᱴ ᱦᱟᱹᱛᱤᱭᱟᱹᱨᱨᱩᱞᱮᱴ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱥᱤᱢᱩᱞᱮᱴᱚᱨᱫᱷᱟᱯ ᱫᱷᱟᱯ ᱥᱮᱥᱚᱱᱫᱷᱟᱯ ᱫᱷᱟᱯ: ᱢᱤᱫ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱥᱮᱥᱚᱱ
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