ꯔꯨꯂꯦꯠꯀꯤ ꯃꯦꯊꯃꯦꯇꯤꯛꯁ · ꯃꯦꯊꯃꯦꯇꯤꯛꯁꯀꯤ ꯁꯃꯁ꯭ꯌꯥ · ꯃꯤꯅꯤꯠ 10

ꯁꯃꯁ꯭ꯌꯥ: ꯇꯦꯕꯜꯒꯤ ꯉꯝꯈꯩ

$500 ꯇꯦꯕꯜ ꯑꯃꯗ $10 ꯒꯤ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ꯫ ꯃꯁꯤ ꯀꯌꯥ ꯌꯥꯝꯅꯥ ꯁꯣꯀꯄꯒꯦ, ꯁꯣꯀꯄꯥ ꯃꯇꯃꯗ ꯀꯌꯥ ꯃꯥꯡꯕꯒꯦ, ꯑꯃꯁꯨꯡ ꯉꯝꯈꯩ ꯋꯥꯡꯍꯟꯕꯅ ꯃꯇꯦꯡ ꯄꯥꯡꯕꯔꯥ?

ꯊꯨꯅ ꯄꯥꯎꯈꯨꯝ

ꯃꯇꯨꯡ ꯏꯟꯅ ꯃꯥꯏꯊꯤꯕ ꯇꯔꯨꯛꯅ ꯃꯁꯤ ꯁꯣꯀꯍꯜꯂꯤ: ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ 1.8%, ꯁ꯭ꯄꯤꯟ 100 ꯒꯤ ꯁꯦꯁꯟꯁꯤꯡꯒꯤ 60% ꯃꯔꯥꯡ ꯀꯥꯏ꯫ ꯁꯣꯀꯄꯅ $630 ꯃꯥꯡꯍꯜꯂꯤ꯫ ꯉꯝꯈꯩ ꯑꯗꯨ $5,000 ꯐꯥꯑꯣꯕ ꯋꯥꯡꯍꯜꯂꯕꯗꯤ ꯁꯣꯀꯄ ꯑꯗꯨ ꯍꯦꯟꯅ ꯈꯟꯅꯔꯛꯏ (ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ 0.25%) ꯑꯃꯁꯨꯡ ꯍꯦꯟꯅ ꯆꯥꯎꯔꯛꯏ ($5,110)꯫ ꯊꯝꯂꯕ ꯗꯣꯂꯥꯔ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯥꯡꯒꯗꯕ ꯁꯦꯜ ꯑꯁꯤ ꯀꯔꯤꯒꯨꯝꯕ ꯉꯝꯈꯩ ꯑꯃꯗꯁꯨ 2.70% ꯑꯣꯏ꯫

ꯑꯍꯧꯕ / ꯉꯝꯈꯩ

$10 / $500

ꯁꯣꯀꯄ ꯐꯥꯑꯣꯕꯒꯤ ꯃꯥꯏꯊꯤꯕ

ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ p(ꯁꯣꯀꯄ)

1.8%

ꯁꯣꯀꯄꯥ ꯃꯇꯃꯒꯤ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ

$630

ꯁꯃꯁ꯭ꯌꯥ

ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯑꯉꯥꯡꯕꯗꯕꯦꯠꯑꯉꯥꯡꯕꯑꯉꯥꯡꯕ ꯃꯁꯤꯡ 18 ꯗ ꯊꯝꯕ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ꯫ ꯐ꯭ꯔꯦꯟꯆ: ꯔꯨꯖ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → $10 ꯊꯝꯂꯤ ꯑꯃꯁꯨꯡ ꯃꯥꯏꯊꯤꯕ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯇꯨꯡꯗ ꯑꯅꯤ ꯊꯣꯛ ꯍꯦꯅꯒꯠꯍꯜꯂꯤ, ꯃꯥꯏꯄꯥꯛꯂꯕꯥ ꯃꯇꯨꯡꯗ ꯑꯍꯧꯕꯗ ꯍꯜꯂꯛꯂꯤ꯫ ꯇꯦꯕꯜꯒꯤ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯑꯁꯤ $500꯫ (a) ꯃꯇꯨꯡ ꯏꯟꯅ ꯃꯥꯏꯊꯤꯕ ꯀꯌꯥꯅ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟꯊꯧꯔꯥꯡꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟꯃꯃꯥꯡꯒꯤ ꯃꯍꯩꯁꯤꯡꯒꯤ ꯃꯈꯥꯗ ꯊꯝꯂꯕ ꯁꯦꯜ ꯍꯣꯡꯍꯟꯕ ꯅꯤꯌꯝ ꯀꯔꯤꯒꯨꯝꯕ꯫ ꯅꯦꯒꯦꯇꯤꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯁꯤꯡꯅ ꯃꯥꯏꯊꯤꯔꯕꯥ ꯃꯇꯨꯡꯗ ꯍꯦꯅꯒꯠꯍꯜꯂꯤ, ꯄꯣꯖꯤꯇꯤꯕꯁꯤꯡꯅ ꯃꯥꯏꯄꯥꯛꯂꯕꯥ ꯃꯇꯨꯡꯗ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯗꯨ ꯁꯣꯀꯍꯜꯂꯤꯕꯒꯦ? (b) ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯃꯦꯊꯃꯦꯇꯤꯛꯁꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯃꯍꯩ ꯑꯃꯥ ꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ, 0 ꯗꯒꯤ 1 ꯐꯥꯑꯣꯕ ꯅꯠꯔꯒ 0% ꯗꯒꯤ 100% ꯐꯥꯑꯣꯕ꯫ ꯃꯁꯤꯡ ꯑꯃꯥ: 1/37 = 2.70%꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯀꯌꯥꯅꯣ? (c) ꯁꯣꯀꯄꯥ ꯃꯇꯃꯗ ꯀꯌꯥ ꯃꯥꯡꯕꯒꯦ? (d) $5,000 ꯒꯤ ꯉꯝꯈꯩ ꯑꯃꯅ ꯄꯥꯎꯈꯨꯃꯁꯤꯡ ꯀꯔꯝꯅ ꯍꯣꯡꯍꯜꯂꯤꯕꯒꯦ? (e) ꯀꯔꯤꯒꯨꯝꯕ ꯉꯝꯈꯩ ꯑꯃꯅ ꯁꯤꯁ꯭ꯇꯦꯝ ꯑꯁꯤ ꯀꯥꯟꯅꯕ ꯑꯣꯏꯍꯜꯂꯤꯕꯔꯥ?

ꯔꯨꯂꯦꯠ ꯇꯦꯕꯜꯒꯤ ꯉꯝꯈꯩꯁꯤꯡ

ꯁꯣꯜꯌꯨꯁꯅꯁꯤꯡ (a)–(c)

ꯊꯝꯂꯕ ꯁꯦꯜꯁꯤꯡ: 10, 20, 40, 80, 160, 320꯫ ꯇꯔꯦꯠꯁꯨꯕ ꯑꯗꯨ 640 > 500 ꯑꯣꯏꯒꯅꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯃꯥꯏꯊꯤꯕ ꯇꯔꯨꯛꯅ ꯃꯁꯤ ꯁꯣꯀꯍꯜꯂꯤ꯫ (b) ꯑꯉꯥꯡꯕꯗ p(ꯃꯥꯏꯊꯤꯕ) = 19/37; (19/37)⁶ = 0.0183, ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ 1.8% ꯃꯔꯥꯡ ꯀꯥꯏ꯫ ꯁꯥꯏꯀꯜ ꯑꯃꯥ ꯑꯦꯚꯔꯦꯖꯗ ꯁ꯭ꯄꯤꯟ ꯑꯅꯤ ꯃꯔꯥꯡ ꯀꯥꯏ ꯂꯩ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯁ꯭ꯄꯤꯟ 100 ꯒꯤ ꯁꯦꯁꯟ ꯑꯃꯗ ꯁꯥꯏꯀꯜ 50 ꯃꯔꯥꯡ ꯀꯥꯏ ꯂꯩ ꯑꯃꯁꯨꯡ p(ꯌꯥꯃꯗ꯭ꯔꯕꯗ ꯁꯣꯀꯄ ꯑꯃꯥ) ≈ 1 − 0.9817⁵⁰ ≈ 60%꯫ (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ꯃꯁꯤ ꯑꯍꯧꯕ ꯌꯨꯅꯤꯠꯁꯦꯜꯌꯨꯅꯤꯠꯁꯤꯁ꯭ꯇꯦꯝ ꯅꯠꯔꯒ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯁꯤꯖꯤꯟꯅꯕ ꯃꯔꯨꯑꯣꯏꯕ ꯊꯝꯂꯕ ꯁꯦꯜ꯫ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯁꯤꯡ ꯌꯨꯅꯤꯠꯇ ꯁꯟꯗꯣꯛꯂꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯃꯈꯣꯌ ꯀꯔꯤꯒꯨꯝꯕ ꯇꯦꯕꯜꯗ ꯆꯠꯂꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → 63 ꯅꯤ, ꯁꯥꯏꯀꯜ ꯑꯗꯨꯅ ꯃꯥꯏꯄꯥꯛꯅꯕ ꯍꯣꯠꯅꯔꯝꯕ $10 ꯒꯤ ꯃꯍꯨꯠꯇ꯫

ꯉꯝꯈꯩꯁꯣꯀꯄ ꯐꯥꯑꯣꯕꯒꯤ ꯃꯥꯏꯊꯤꯕꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ p(ꯁꯣꯀꯄ)ꯁꯣꯀꯄꯥ ꯃꯇꯃꯒꯤ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜp(ꯁ꯭ꯄꯤꯟ 100 ꯗ ꯁꯣꯀꯄ)ꯍꯟꯅ ꯐꯪꯅꯕ ꯃꯊꯧ ꯇꯥꯕ ꯃꯥꯏꯄꯥꯛꯄ

$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

ꯉꯝꯈꯩ ꯑꯗꯨ ꯑꯅꯤ ꯊꯣꯛ ꯍꯦꯅꯒꯠꯍꯅꯕ ꯈꯨꯗꯤꯡꯗ ꯁꯣꯀꯄ ꯑꯗꯨ ꯇꯉꯥꯏ ꯃꯔꯥꯡ ꯀꯥꯏꯅ ꯍꯟꯅ ꯊꯣꯀꯏ ꯑꯃꯁꯨꯡ ꯃꯃꯜ ꯑꯅꯤ ꯊꯣꯛ ꯆꯥꯡꯏ꯫ ꯃꯈꯣꯌꯒꯤ ꯃꯜꯇꯤꯞꯂꯥꯏ ꯑꯗꯨ ꯌꯥꯝꯅꯥ ꯍꯣꯡꯗꯦ꯫

ꯁꯣꯜꯌꯨꯁꯟ (d)

$5,000 ꯗ: 2,560 ꯐꯥꯑꯣꯕ ꯊꯝꯕ ꯌꯥꯏ; ꯇꯔꯥꯁꯨꯕ ꯊꯝꯂꯕ ꯁꯦꯜ, 5,120, ꯌꯥꯗꯦ꯫ ꯃꯥꯏꯊꯤꯕ ꯃꯥꯄꯟꯅ ꯃꯁꯤ ꯁꯣꯀꯍꯜꯂꯤ: (19/37)⁹ ≈ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ 0.25%, ꯁ꯭ꯄꯤꯟ 100 ꯒꯤ ꯁꯦꯁꯟꯁꯤꯡꯒꯤ 11% ꯃꯔꯥꯡ ꯀꯥꯏ꯫ ꯁꯣꯀꯄꯥ ꯃꯇꯃꯒꯤ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ: 10 + 20 + … + 2,560 = $5,110꯫ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯗꯨꯅ ꯍꯦꯟꯅ ꯆꯥꯎꯕ ꯑꯋꯥꯕꯒꯤ ꯃꯃꯂꯗ ꯍꯦꯟꯅ ꯈꯟꯅꯕ ꯑꯋꯥꯕ ꯑꯃꯥ ꯂꯩꯔꯦ꯫ ꯁꯦꯁꯟ ꯀꯌꯥꯗ ꯁꯣꯀꯄꯁꯤꯡꯒꯤ ꯃꯥꯡꯒꯗꯕ ꯃꯃꯜ ꯑꯁꯤ p × ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ ≈ $500 ꯉꯝꯈꯩꯗ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ 0.0183 × 630 = $11.5 ꯑꯃꯁꯨꯡ $5,000 ꯗ 0.0025 × 5,110 = $12.8; ꯑꯅꯤ ꯑꯁꯤ ꯅꯛꯅꯩ ꯑꯗꯨꯕꯨ ꯆꯞ ꯃꯥꯟꯅꯗꯦ, ꯑꯃꯁꯨꯡ ꯃꯈꯣꯌ ꯈꯨꯗꯤꯡꯃꯛ ꯁꯥꯏꯀꯜ ꯑꯗꯨꯒꯤ ꯃꯥꯡꯒꯗꯕ ꯁꯦꯜꯃꯦꯊꯃꯦꯇꯤꯛꯁꯃꯥꯡꯒꯗꯕ ꯁꯦꯜꯐꯪꯒꯗꯕ ꯃꯃꯂꯕꯨ ꯕꯦꯠ ꯊꯝꯂꯕ ꯁꯦꯜ ꯄꯨꯝꯅꯃꯛꯅ ꯃꯅꯥꯡꯕ; ꯁꯦꯁꯟ ꯅꯠꯔꯒ ꯃꯇꯝ ꯑꯃꯒꯤ ꯃꯅꯨꯡꯗ ꯃꯥꯡꯕ ꯑꯦꯚꯔꯦꯖ ꯁꯦꯜ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯁꯨꯡ ꯃꯥꯏꯄꯥꯛꯂꯕ ꯁꯥꯏꯀꯂꯁꯤꯡꯅ ꯐꯪꯂꯤꯕ $10 ꯄꯨꯟꯁꯤꯅꯕꯅꯤ꯫

ꯁꯣꯜꯌꯨꯁꯟ (e)

ꯅꯠꯇꯦ꯫ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯥꯡꯒꯗꯕ ꯁꯦꯜ = (ꯁꯥꯏꯀꯜ ꯑꯗꯨꯗ ꯊꯝꯒꯗꯕ ꯄꯨꯝꯅꯃꯛ ꯁꯦꯜ) / 37, ꯀꯔꯤꯒꯨꯝꯕ ꯉꯝꯈꯩ ꯑꯃꯗꯁꯨ, ꯃꯔꯃꯗꯤ ꯕꯦꯠ ꯈꯨꯗꯤꯡꯃꯛ ꯃꯁꯥꯒꯤ −1/37 ꯑꯣꯏ꯫ ꯉꯝꯈꯩ ꯑꯗꯨꯅ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ ꯑꯗꯨ ꯑꯄꯤꯛꯄ ꯃꯥꯏꯄꯥꯛꯄ ꯀꯌꯥ ꯑꯃꯁꯨꯡ ꯑꯆꯧꯕ ꯃꯥꯏꯊꯤꯕ ꯈꯔꯒꯤ ꯃꯔꯛꯇ ꯀꯔꯝꯅ ꯌꯦꯟꯊꯣꯛꯀꯅꯤ ꯍꯥꯌꯕ ꯈꯛꯇ ꯂꯦꯞꯊꯣꯀꯏ꯫ ꯉꯝꯈꯩ ꯂꯩꯇꯕ ꯑꯃꯁꯨꯡ ꯂꯣꯏꯕ ꯅꯥꯏꯗꯕ ꯕꯦꯡꯀꯔꯣꯜꯁꯦꯜꯕꯦꯡꯀꯔꯣꯜꯁꯥꯟꯅꯅꯕ ꯇꯣꯉꯥꯟꯅ ꯊꯝꯂꯕ ꯁꯦꯜ ꯄꯨꯝꯅꯃꯛ꯫ ꯕꯦꯡꯀꯔꯣꯜ ꯃꯦꯅꯦꯖꯃꯦꯟꯠ ꯍꯥꯌꯕꯁꯤ ꯃꯗꯨꯒꯤ ꯀꯌꯥ ꯕꯦꯠ ꯈꯨꯗꯤꯡꯗ ꯑꯃꯁꯨꯡ ꯁꯦꯁꯟ ꯈꯨꯗꯤꯡꯗ ꯈꯨꯗꯣꯡꯊꯤꯕꯗ ꯊꯃꯒꯗꯒꯦ ꯍꯥꯌꯕ ꯋꯥꯔꯦꯞ ꯂꯧꯕꯅꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯂꯩꯔꯕꯗꯤ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ ꯑꯁꯤ “ꯊꯕꯛ ꯄꯥꯡꯊꯣꯛꯀꯅꯤ”, ꯀꯩꯗꯧꯉꯩꯗꯁꯨ ꯁꯣꯛꯂꯣꯏꯗꯕ ꯍꯥꯌꯕ ꯑꯔꯊꯗ, ꯑꯃꯁꯨꯡ ꯃꯁꯤꯅ ꯉꯝꯈꯩ ꯂꯩꯇꯕ ꯁꯦꯜ ꯊꯝꯒꯅꯤ, ꯃꯗꯨꯒꯤ 2.70% ꯃꯥꯡꯒꯗꯕ ꯁꯦꯜ ꯑꯣꯏꯅ꯫ ꯃꯥꯔꯇꯤꯟꯒꯦꯜꯊꯧꯔꯥꯡꯃꯥꯔꯇꯤꯟꯒꯦꯜꯃꯥꯏꯊꯤꯕ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯇꯨꯡꯗ ꯕꯦꯠ ꯑꯗꯨ ꯑꯅꯤ ꯊꯥꯛ ꯍꯦꯅꯒꯠꯍꯟꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ꯫ ꯃꯁꯤꯅ ꯀꯔꯤ ꯇꯧꯕꯒꯦ ꯑꯃꯁꯨꯡ ꯀꯔꯤ ꯇꯧꯗꯕꯒꯦ ꯍꯥꯌꯕ ꯊꯧꯔꯥꯡꯒꯤ ꯁꯦꯛꯁꯅꯗ ꯌꯦꯡꯕꯤꯌꯨ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯊꯕꯛ ꯄꯥꯡꯊꯣꯀꯄꯗꯒꯤ ꯊꯤꯡꯂꯤꯕꯗꯤ ꯇꯦꯕꯜꯒꯤ ꯉꯝꯈꯩꯇꯦꯕꯜꯇꯦꯕꯜꯒꯤ ꯉꯝꯈꯩꯌꯥꯍꯜꯂꯤꯕ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯅꯦꯝꯕ ꯑꯃꯁꯨꯡ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯕꯦꯠ꯫ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯑꯗꯨꯅ ꯑꯅꯤ ꯊꯥꯛ ꯍꯦꯅꯒꯠꯍꯟꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯁꯤꯡ ꯊꯤꯡꯏ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯅꯠꯇꯦ꯫ ꯃꯁꯤꯡ ꯊꯤꯕꯅꯤ꯫

$10 ꯑꯍꯧꯕꯗ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ, ꯁ꯭ꯄꯤꯟ 100: ꯉꯝꯈꯩꯒꯤ ꯃꯇꯨꯡ ꯏꯟꯅ ꯑꯦꯚꯔꯦꯖ ꯃꯍꯩ (ꯁꯤꯃꯨꯂꯦꯠ ꯇꯧꯔꯕ)$

$500 ꯉꯝꯈꯩ

≈ −$75

$1,000 ꯉꯝꯈꯩ

≈ −$75

$2,000 ꯉꯝꯈꯩ

≈ −$72

$5,000 ꯉꯝꯈꯩ

≈ −$70

ꯐ꯭ꯂꯦꯠ $10, ꯀꯔꯤꯒꯨꯝꯕ ꯉꯝꯈꯩ

$27

04080

ꯕꯥꯔꯁꯤꯡꯅ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜꯒꯤ ꯆꯥꯡ ꯎꯠꯂꯤ꯫ ꯉꯝꯈꯩ ꯋꯥꯡꯍꯟꯕꯅ ꯑꯦꯚꯔꯦꯖ ꯑꯗꯨ ꯌꯥꯝꯅꯥ ꯍꯣꯡꯍꯜꯂꯤꯗꯦ; ꯉꯝꯈꯩ ꯀꯔꯝꯕ ꯑꯣꯏꯔꯕꯁꯨ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ ꯑꯁꯤꯅ ꯁ꯭ꯄꯤꯟ 100 ꯈꯨꯗꯤꯡꯗ $2,800 ꯃꯔꯥꯡ ꯀꯥꯏ ꯊꯝꯂꯤ꯫

ꯑꯁꯦꯡꯕ ꯇꯦꯕꯜ ꯑꯅꯤ

ꯊꯣꯔꯄꯀꯤ ꯈꯨꯗꯝ ꯑꯁꯤ $1,000 ꯒꯤ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯂꯩꯕ $1 ꯒꯤ ꯇꯦꯕꯜ ꯑꯃꯅꯤ, ꯃꯐꯝ ꯑꯗꯨꯗ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯅꯨꯃꯤꯗꯥꯡꯒꯤ ꯆꯥꯛꯀꯤꯗꯃꯛ $30 ꯗꯣꯂꯥꯔ ꯑꯃꯃꯃꯛ ꯃꯥꯏꯄꯥꯛꯅꯕ ꯑꯥꯁꯥ ꯇꯧꯔꯤ꯫ ꯊꯝꯂꯕ ꯁꯦꯜ 1, 2, 4 … 512 ꯅ ꯕꯦꯠ ꯇꯔꯥ ꯌꯥꯍꯜꯂꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯃꯇꯨꯡ ꯏꯟꯅ ꯃꯥꯏꯊꯤꯕ ꯇꯔꯥꯅ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ ꯑꯗꯨ ꯁꯣꯀꯍꯜꯂꯤ: ꯌꯨꯔꯣꯄꯤꯌꯟ ꯆꯛꯔ ꯑꯃꯗ ꯁꯥꯏꯀꯜ 785 ꯗ ꯑꯃꯨꯛ (ꯑꯃꯦꯔꯤꯀꯟ ꯑꯃꯗ 613 ꯗ ꯑꯃꯨꯛ), $1,023 ꯒꯤ ꯃꯃꯂꯗ꯫ ꯁꯥꯏꯀꯜ ꯀꯨꯟꯊ꯭ꯔꯥ ꯄꯨꯝꯅꯃꯛ ꯃꯇꯃꯒꯤ 96.2% ꯗ ꯃꯥꯏꯄꯥꯛꯏ꯫ ꯑꯇꯣꯞꯄ 3.8% ꯅ $1,000 ꯃꯔꯥꯡ ꯀꯥꯏ ꯃꯥꯡꯏ, ꯑꯃꯁꯨꯡ ꯅꯨꯃꯤꯗꯥꯡ ꯑꯗꯨꯒꯤ ꯑꯦꯚꯔꯦꯖ ꯃꯍꯩ −$9.17 ꯑꯣꯏ, ꯃꯁꯤ ꯆꯨꯝꯅ ꯁꯥꯏꯀꯜ ꯀꯨꯟꯊ꯭ꯔꯥꯅ ꯑꯦꯚꯔꯦꯖꯗ ꯊꯝꯂꯤꯕ $339 ꯒꯤ 1/37 ꯅꯤ꯫ $2,000 ꯒꯤ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯂꯩꯕ $5 ꯒꯤ ꯇꯦꯕꯜ ꯑꯃꯁꯨ ꯈꯦꯠꯅꯗꯦ: $1,280 ꯐꯥꯑꯣꯕꯒꯤ ꯊꯝꯂꯕ ꯁꯦꯜꯅ ꯕꯦꯠ ꯃꯥꯄꯟ ꯌꯥꯍꯜꯂꯤ, ꯑꯃꯁꯨꯡ ꯁꯣꯀꯄ ꯑꯁꯤ ꯌꯨꯔꯣꯄꯤꯌꯟ ꯆꯛꯔ ꯑꯃꯗ ꯁꯥꯏꯀꯜ 403 ꯗ ꯑꯃꯨꯛ (ꯑꯃꯦꯔꯤꯀꯟ ꯑꯃꯗ 323 ꯗ ꯑꯃꯨꯛ) ꯂꯥꯀꯏ ꯑꯃꯁꯨꯡ $2,555 ꯃꯥꯡꯍꯜꯂꯤ꯫

ꯑꯅꯤ ꯊꯣꯛ ꯍꯦꯅꯒꯠꯂꯤꯕ ꯊꯝꯂꯕ ꯁꯦꯜꯁꯤꯡ ꯇꯦꯕꯜꯒꯤ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯉꯝꯈꯩꯗ ꯆꯠꯂꯤ, ꯑꯗꯨꯒ ꯅꯨꯃꯤꯗꯥꯡꯒꯤ ꯆꯥꯛ ꯃꯅꯥꯛꯀꯤ ꯀꯥꯗ ꯉꯥꯏꯔꯤ꯫

ꯁꯦꯜ ꯌꯥꯝꯅꯥ ꯂꯩꯕ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯥ

ꯉꯝꯈꯩ ꯅꯦꯝꯕꯒꯤ ꯃꯍꯨꯠꯇ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ ꯑꯗꯨꯗ ꯌꯥꯝꯅꯥ ꯆꯥꯎꯕ ꯕꯦꯡꯀꯔꯣꯜ ꯑꯃꯥ ꯄꯤꯁꯤ: ꯌꯨꯅꯤꯠ 1,000, ꯌꯨꯅꯤꯠ ꯑꯃꯒꯤ ꯑꯍꯧꯕ ꯑꯃꯁꯨꯡ ꯑꯉꯥꯡꯕꯗ ꯁ꯭ꯄꯤꯟ 1,000, ꯃꯊꯪꯒꯤ ꯑꯅꯤ ꯊꯣꯛ ꯑꯗꯨ ꯂꯦꯃꯍꯧꯔꯤꯕꯗꯒꯤ ꯍꯦꯅꯕ ꯃꯇꯝ ꯈꯨꯗꯤꯡꯗ ꯌꯨꯅꯤꯠ ꯑꯃꯗꯒꯤ ꯑꯃꯨꯛ ꯍꯟꯅ ꯍꯧꯕ꯫ ꯑꯩꯈꯣꯌꯅ ꯑꯁꯤꯒꯨꯝꯕ ꯅꯨꯃꯤꯗꯥꯡ ꯂꯥꯈ ꯇꯔꯥ ꯁꯤꯃꯨꯂꯦꯠ ꯇꯧꯈꯤ꯫ ꯌꯨꯔꯣꯄꯤꯌꯟ ꯆꯛꯔ ꯑꯃꯗ 52% ꯅ ꯀꯥꯟꯅꯕ ꯐꯪꯗꯨꯅ ꯂꯣꯏꯏ, 21% ꯅ ꯄꯨꯝꯅꯃꯛ ꯃꯥꯡꯏ, ꯑꯃꯁꯨꯡ ꯑꯦꯚꯔꯦꯖ ꯃꯍꯩ ꯑꯁꯤ ꯌꯨꯅꯤꯠ −124 ꯑꯣꯏ꯫ ꯕꯦꯠ ꯈꯨꯗꯤꯡꯃꯛ ꯆꯨꯝꯃꯤꯟꯅꯕ, ꯖꯤꯔꯣꯆꯛꯔꯖꯤꯔꯣꯆꯞ ꯃꯥꯟꯅꯕ, ꯗꯖꯟ ꯅꯠꯔꯒ ꯀꯂꯝ ꯕꯦꯠ ꯑꯃꯠꯇꯅ ꯀꯨꯞꯇꯕ ꯑꯁꯪꯕ ꯄꯣꯀꯦꯠ ꯑꯃꯈꯛꯇ; ꯍꯥꯎꯁ ꯑꯦꯖꯀꯤ ꯍꯧꯔꯀꯐꯝꯅꯤ꯫ ꯑꯇꯣꯞꯄ ꯃꯃꯤꯡ: ꯁꯤꯡꯒꯜ ꯖꯤꯔꯣ, 0꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯂꯩꯇꯕ ꯁꯥꯟꯅꯕ ꯑꯃꯗꯁꯨ 60% ꯃꯔꯥꯡ ꯀꯥꯏ ꯈꯛꯇꯅ ꯀꯥꯟꯅꯕ ꯐꯪꯗꯨꯅ ꯂꯣꯏꯒꯅꯤ꯫ ꯃꯤꯅꯝ ꯅꯨꯃꯤꯗꯥꯡ ꯑꯁꯤ ꯑꯄꯤꯛꯄ ꯃꯥꯏꯄꯥꯛꯄ ꯑꯃꯅꯤ; ꯑꯦꯚꯔꯦꯖ ꯑꯗꯨꯒꯤ ꯃꯃꯜ ꯄꯤꯔꯤꯕꯗꯤ ꯖꯤꯔꯣꯗ ꯂꯣꯏꯔꯤꯕ ꯅꯨꯃꯤꯗꯥꯡꯁꯤꯡ ꯑꯗꯨꯅꯤ꯫

ꯆꯛꯔꯀꯥꯟꯅꯕ ꯐꯪꯗꯨꯅ ꯂꯣꯏꯕꯌꯨꯅꯤꯠ 1,000 ꯄꯨꯝꯅꯃꯛ ꯃꯥꯡꯕꯑꯦꯚꯔꯦꯖ ꯃꯍꯩ

ꯖꯤꯔꯣ ꯂꯩꯇꯕ (ꯆꯨꯝꯃꯤꯟꯅꯕ ꯁꯥꯟꯅꯕ)60%15%0

ꯌꯨꯔꯣꯄꯤꯌꯟ52%21%ꯌꯨꯅꯤꯠ −124

ꯑꯃꯦꯔꯤꯀꯟ45%30%ꯌꯨꯅꯤꯠ −250

ꯑꯩꯈꯣꯌꯒꯤ ꯁꯤꯃꯨꯂꯦꯁꯟ, ꯆꯛꯔ ꯈꯨꯗꯤꯡꯗ ꯅꯨꯃꯤꯗꯥꯡ ꯂꯥꯈ ꯇꯔꯥ꯫ ꯑꯦꯚꯔꯦꯖꯁꯤꯡ ꯑꯁꯤ ꯁ꯭ꯄꯤꯟ 1,000 ꯗ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯅ ꯊꯝꯂꯤꯕ ꯁꯦꯜꯒꯤ 1/37 ꯑꯃꯁꯨꯡ 1/19 ꯅꯤ: ꯌꯨꯔꯣꯄꯤꯌꯟ ꯆꯛꯔ ꯑꯃꯗ ꯌꯨꯅꯤꯠ 4,600 ꯃꯔꯥꯡ ꯀꯥꯏ ꯑꯃꯁꯨꯡ ꯑꯃꯦꯔꯤꯀꯟ ꯑꯃꯗ 4,750 ꯃꯔꯥꯡ ꯀꯥꯏ꯫

ꯇꯃꯖꯕ ꯋꯥꯐꯝ

ꯀꯦꯁꯤꯅꯣꯁꯤꯡꯅ ꯉꯝꯈꯩ ꯊꯝꯕ ꯑꯁꯤ ꯃꯈꯣꯌ ꯃꯁꯥꯒꯤ ꯚꯦꯌꯔꯤꯌꯦꯟꯁꯃꯦꯊꯃꯦꯇꯤꯛꯁꯚꯦꯌꯔꯤꯌꯦꯟꯁꯃꯍꯩꯁꯤꯡ ꯐꯪꯒꯗꯕ ꯃꯃꯂꯒꯤ ꯑꯀꯣꯌꯕꯗ ꯀꯌꯥ ꯁꯟꯗꯣꯛꯅ ꯂꯩꯕꯒꯦ ꯍꯥꯌꯕꯒꯤ ꯆꯥꯡ꯫ ꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯇ ꯋꯥꯡꯏ, ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠꯁꯤꯡꯗ ꯅꯦꯝꯏ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯉꯝꯈꯩ ꯊꯝꯅꯕꯅꯤ, ꯁꯤꯁ꯭ꯇꯦꯝꯁꯤꯡ ꯃꯥꯏꯊꯤꯍꯟꯅꯕ ꯅꯠꯇꯦ꯫ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ ꯃꯥꯏꯊꯤꯍꯟꯕ ꯑꯁꯤ ꯃꯈꯣꯌꯅ ꯅꯨꯡꯉꯥꯏꯅ ꯂꯧꯕ ꯁꯥꯏꯗ ꯏꯐꯦꯛꯇ ꯑꯃꯅꯤ, ꯑꯃꯁꯨꯡ ꯃꯁꯤꯡ ꯊꯤꯕꯅ ꯃꯁꯤ ꯀꯔꯝꯕ ꯑꯣꯏꯔꯕꯁꯨ ꯃꯥꯏꯊꯤꯍꯅꯒꯅꯤ꯫ ꯉꯝꯈꯩ ꯑꯗꯨꯅꯤ ꯁꯃꯁ꯭ꯌꯥ ꯍꯥꯌꯅ ꯈꯜꯂꯤꯕ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯆꯤꯡꯒꯤ ꯇꯨꯝꯕ ꯃꯐꯝ ꯑꯗꯨ ꯊꯤꯗꯣꯀꯈ꯭ꯔꯦ ꯑꯗꯨꯕꯨ ꯃꯗꯨꯕꯨ ꯆꯨꯃꯊꯣꯛ ꯍꯥꯌꯅ ꯑꯔꯥꯟꯅ ꯂꯧꯔꯦ꯫

ꯐꯣꯔꯃꯨꯂꯥ

p(ꯁꯣꯀꯄ) = (19/37)ᵏ, k = ꯉꯝꯈꯩꯅ ꯌꯥꯍꯜꯂꯤꯕ ꯃꯥꯏꯊꯤꯕ

ꯁꯣꯀꯄꯥ ꯃꯇꯃꯒꯤ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ = ꯑꯍꯧꯕ · (2ᵏ − 1)

E[ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯥꯡꯈꯤꯕ ꯁꯦꯜ] = E[ꯁꯥꯏꯀꯂꯗ ꯊꯝꯂꯕ ꯁꯦꯜ] / 37

ꯋꯥꯍꯪꯁꯤꯡ

ꯆꯥꯎꯕ ꯕꯦꯡꯀꯔꯣꯜ ꯑꯃꯅ ꯃꯁꯤ ꯁꯦꯃꯒꯠꯀꯗ꯭ꯔꯥ?

ꯃꯁꯤꯅ ꯅꯍꯥꯀꯄꯨ ꯍꯦꯟꯅ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯐꯥꯑꯣꯕ ꯍꯤꯡꯍꯜꯂꯤ, ꯃꯐꯝ ꯑꯗꯨꯗ ꯁꯣꯀꯄ ꯑꯁꯤ ꯍꯦꯟꯅ ꯈꯟꯅꯩ ꯑꯃꯁꯨꯡ ꯍꯦꯟꯅ ꯆꯥꯎꯏ꯫ ꯗꯣꯂꯥꯔ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯥꯡꯒꯗꯕ ꯁꯦꯜ ꯍꯣꯡꯗꯦ꯫

ꯑꯅꯤ ꯊꯣꯛ ꯍꯦꯅꯒꯠꯍꯅꯕ ꯈꯨꯗꯤꯡꯗ ꯌꯨꯅꯤꯠ ꯑꯃꯥ ꯍꯥꯄꯆꯤꯟꯕ “ꯒ꯭ꯔꯦꯟꯗ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ” ꯗꯤ ꯀꯔꯝꯅꯣ?

ꯊꯝꯂꯕ ꯁꯦꯜ 10, 30, 70, 150, 310: ꯃꯥꯏꯊꯤꯕ ꯃꯉꯥꯒꯤ ꯃꯇꯨꯡꯗ $500 ꯉꯝꯈꯩ ꯑꯗꯨ ꯌꯧꯏ, ꯑꯃꯁꯨꯡ ꯁꯣꯀꯄꯅ $570 ꯃꯥꯡꯍꯜꯂꯤ꯫ 2.70% ꯑꯗꯨꯃꯛ꯫

ꯀꯦꯁꯤꯅꯣꯁꯤꯡꯗ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯀꯔꯤꯒꯤ ꯂꯩꯕꯒꯦ?

ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯀꯦꯁꯤꯅꯣꯒꯤ ꯊꯝꯂꯕ ꯁꯦꯜꯅ ꯂꯥꯏꯅ ꯄꯤꯕ ꯉꯝꯕꯗꯒꯤ ꯍꯦꯟꯅ ꯕꯦꯠ ꯊꯝꯕ ꯉꯃꯗꯅꯕ꯫ ꯚꯦꯌꯔꯤꯌꯦꯟꯁ ꯃꯦꯅꯦꯖꯃꯦꯟꯠ꯫

ꯃꯦꯊꯃꯦꯇꯤꯛꯁꯀꯥ ꯁꯥꯟꯅꯕ

ꯃꯁꯤꯁꯤꯡ ꯑꯍꯥꯅꯕꯗ ꯅꯍꯥꯛ ꯃꯁꯥꯃꯛꯅ ꯍꯣꯠꯅꯕꯤꯌꯨ꯫ ꯄꯥꯎꯈꯨꯝ ꯈꯨꯗꯤꯡꯃꯛ ꯊꯨꯡꯗꯨꯅ ꯊꯝꯂꯤ; ꯁꯦꯝ ꯁꯥꯔꯕ ꯃꯇꯃꯗ ꯍꯥꯡꯗꯣꯛꯄꯤꯌꯨ꯫

ꯃꯥꯔꯇꯤꯟꯒꯦꯜꯊꯧꯔꯥꯡꯃꯥꯔꯇꯤꯟꯒꯦꯜꯃꯥꯏꯊꯤꯕ ꯈꯨꯗꯤꯡꯒꯤ ꯃꯇꯨꯡꯗ ꯕꯦꯠ ꯑꯗꯨ ꯑꯅꯤ ꯊꯥꯛ ꯍꯦꯅꯒꯠꯍꯟꯕ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯟ꯫ ꯃꯁꯤꯅ ꯀꯔꯤ ꯇꯧꯕꯒꯦ ꯑꯃꯁꯨꯡ ꯀꯔꯤ ꯇꯧꯗꯕꯒꯦ ꯍꯥꯌꯕ ꯊꯧꯔꯥꯡꯒꯤ ꯁꯦꯛꯁꯅꯗ ꯌꯦꯡꯕꯤꯌꯨ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯅ ꯑꯉꯥꯡꯕꯕꯦꯠꯑꯉꯥꯡꯕꯑꯉꯥꯡꯕ ꯃꯁꯤꯡ 18 ꯗ ꯊꯝꯕ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ꯫ ꯐ꯭ꯔꯦꯟꯆ: ꯔꯨꯖ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯗ $100 ꯗꯒꯤ ꯍꯧꯏ ꯑꯃꯁꯨꯡ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ ꯕꯦꯠ n ꯐꯥꯑꯣꯕꯈꯛ ꯌꯥꯍꯜꯂꯤ: ꯃꯈꯥ ꯆꯠꯅꯅ ꯃꯥꯏꯊꯤꯕ n ꯒꯤ ꯃꯇꯨꯡꯗ ꯊꯥꯗꯣꯛꯇꯨꯅ ꯑꯅꯧꯅ ꯍꯧꯏ꯫ ꯁꯥꯏꯀꯜ ꯑꯃꯒꯤ ꯑꯦꯚꯔꯦꯖ ꯃꯍꯩ ꯑꯗꯨ n ꯗ ꯐꯣꯔꯃꯨꯂꯥ ꯑꯃꯥ ꯑꯣꯏꯅ ꯏꯕꯤꯌꯨ, ꯑꯃꯁꯨꯡ ꯌꯨꯔꯣꯄꯤꯌꯟ ꯆꯛꯔꯗ ꯕꯦꯠ 5, 6 ꯑꯃꯁꯨꯡ 7 ꯒꯤꯗꯃꯛ ꯍꯤꯁꯥꯕ ꯇꯧꯕꯤꯌꯨ꯫
ꯄꯥꯎꯈꯨꯝ ꯎꯠꯄ
ꯄꯥꯎꯈꯨꯝ ꯑꯦꯚꯔꯦꯖ ꯑꯁꯤ 100 × (1 − (38/37)ⁿ): ꯕꯦꯠ 5 ꯒꯤꯗꯃꯛ −$14.26, 6 ꯒꯤꯗꯃꯛ −$17.35 ꯑꯃꯁꯨꯡ 7 ꯒꯤꯗꯃꯛ −$20.52꯫ ꯕꯦꯠ ꯍꯦꯟꯅ ꯌꯥꯍꯜꯂꯕꯅ ꯑꯦꯚꯔꯦꯖ ꯁꯥꯏꯀꯜ ꯑꯗꯨ ꯍꯦꯟꯅ ꯐꯠꯍꯜꯂꯤ, ꯐꯒꯠꯍꯜꯂꯗꯦ꯫
ꯕꯦꯠ n ꯄꯨꯝꯅꯃꯛ ꯃꯥꯏꯊꯤꯕꯗ꯭ꯔꯕꯗꯤ ꯁꯥꯏꯀꯜ ꯑꯃꯅ $100 ꯃꯥꯏꯄꯥꯀꯏ, ꯃꯁꯤ qⁿ ꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯒꯥ ꯂꯣꯌꯅꯅ ꯊꯣꯛꯏ, ꯃꯐꯝ ꯑꯗꯨꯗ q = 19/37꯫ ꯄꯨꯝꯅꯃꯛ ꯃꯥꯏꯊꯤꯕꯔꯕꯗꯤ, ꯊꯝꯂꯕ ꯁꯦꯜ 100, 200 … 100 × 2ⁿ⁻¹ ꯄꯨꯟꯁꯤꯜꯂꯕꯗ 100 × (2ⁿ − 1) ꯑꯣꯏ꯫ ꯃꯔꯝ ꯑꯗꯨꯅ ꯑꯦꯚꯔꯦꯖ ꯑꯁꯤ 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ)꯫
ꯃꯐꯝ ꯑꯁꯤꯗ 2q = 38/37, 1 ꯗꯒꯤ ꯈꯔꯥ ꯍꯦꯟꯅ, ꯃꯔꯝ ꯑꯗꯨꯅ (2q)ⁿ ꯑꯁꯤ n ꯒꯥ ꯂꯣꯌꯅꯅ ꯍꯦꯅꯒꯠꯂꯀꯏ ꯑꯃꯁꯨꯡ ꯑꯦꯚꯔꯦꯖ ꯑꯗꯨ ꯇꯥꯔꯛꯏ: n = 5 ꯅ 1.1426 ꯑꯃꯁꯨꯡ −$14.26 ꯄꯤ; n = 6 ꯅ 1.1735 ꯑꯃꯁꯨꯡ −$17.35 ꯄꯤ; n = 7 ꯅ 1.2052 ꯑꯃꯁꯨꯡ −$20.52 ꯄꯤ꯫ ꯍꯦꯟꯅ ꯁꯥꯡꯕ ꯈꯣꯡꯂꯝ ꯑꯃꯅ ꯍꯦꯟꯅ ꯈꯟꯅ ꯃꯥꯡꯏ, ꯁꯥꯏꯀꯂꯒꯤ 3.6%, 1.8% ꯑꯃꯁꯨꯡ 0.9% ꯗ, ꯑꯗꯨꯕꯨ ꯃꯥꯡꯕ ꯈꯨꯗꯤꯡ ꯑꯃꯨꯛ ꯑꯅꯤ ꯆꯥꯎꯏ, ꯑꯃꯁꯨꯡ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ ꯇꯦꯕꯜ ꯃꯊꯛꯇ ꯁꯦꯜ ꯍꯦꯟꯅ ꯆꯠꯏ, ꯕꯦꯠ ꯈꯨꯗꯤꯡꯒꯤ 2.70% ꯗ꯫
ꯑꯃꯦꯔꯤꯀꯟ ꯆꯛꯔꯗ 2q = 40/38 ꯑꯃꯁꯨꯡ ꯃꯁꯤꯡꯁꯤꯡ ꯑꯗꯨ −$29.24, −$36.04 ꯑꯃꯁꯨꯡ −$43.20 (Taylor, 2021)꯫ ꯖꯤꯔꯣꯆꯛꯔꯖꯤꯔꯣꯆꯞ ꯃꯥꯟꯅꯕ, ꯗꯖꯟ ꯅꯠꯔꯒ ꯀꯂꯝ ꯕꯦꯠ ꯑꯃꯠꯇꯅ ꯀꯨꯞꯇꯕ ꯑꯁꯪꯕ ꯄꯣꯀꯦꯠ ꯑꯃꯈꯛꯇ; ꯍꯥꯎꯁ ꯑꯦꯖꯀꯤ ꯍꯧꯔꯀꯐꯝꯅꯤ꯫ ꯑꯇꯣꯞꯄ ꯃꯃꯤꯡ: ꯁꯤꯡꯒꯜ ꯖꯤꯔꯣ, 0꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯂꯩꯇꯕ ꯁꯥꯟꯅꯕ ꯑꯃꯗ 2q = 1, ꯑꯃꯁꯨꯡ ꯈꯣꯡꯂꯝ ꯑꯗꯨ ꯀꯌꯥ ꯁꯥꯡꯂꯕꯁꯨ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯒꯤ ꯑꯦꯚꯔꯦꯖ ꯆꯨꯝꯅ ꯖꯤꯔꯣ꯫
“ꯒ꯭ꯔꯦꯟꯗ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ” ꯅ ꯃꯥꯏꯊꯤꯕ ꯑꯃꯒꯤ ꯃꯇꯨꯡꯗ ꯊꯝꯂꯕ ꯁꯦꯜ ꯑꯃꯨꯛ ꯑꯅꯤ ꯇꯧꯗꯨꯅ ꯌꯨꯅꯤꯠꯁꯦꯜꯌꯨꯅꯤꯠꯁꯤꯁ꯭ꯇꯦꯝ ꯅꯠꯔꯒ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯁꯤꯖꯤꯟꯅꯕ ꯃꯔꯨꯑꯣꯏꯕ ꯊꯝꯂꯕ ꯁꯦꯜ꯫ ꯄ꯭ꯔꯣꯒ꯭ꯔꯦꯁꯅꯁꯤꯡ ꯌꯨꯅꯤꯠꯇ ꯁꯟꯗꯣꯛꯂꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯃꯈꯣꯌ ꯀꯔꯤꯒꯨꯝꯕ ꯇꯦꯕꯜꯗ ꯆꯠꯂꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯥ ꯍꯥꯄꯆꯤꯜꯂꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯊꯝꯂꯕ ꯁꯦꯜ 1, 3, 7, 15, 31 … ꯑꯣꯏ꯫ ꯑꯍꯥꯅꯕ ꯃꯥꯏꯄꯥꯛꯄ ꯑꯗꯨ 1 ꯁꯨꯕ, 2 ꯁꯨꯕ, 3 ꯁꯨꯕ ꯅꯠꯔꯒꯥ k ꯁꯨꯕ ꯕꯦꯠꯇ ꯂꯥꯛꯂꯕꯗꯤ ꯀꯌꯥ ꯀꯥꯟꯅꯒꯅꯤ? $10 ꯒꯤ ꯌꯨꯅꯤꯠ ꯑꯃꯁꯨꯡ $500 ꯒꯤ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕꯗ, ꯁꯥꯏꯀꯜ ꯑꯃꯥ ꯀꯌꯥ ꯌꯥꯃꯅ ꯁꯣꯛꯀꯅꯤ, ꯑꯃꯁꯨꯡ ꯁꯥꯏꯀꯜ ꯑꯃꯅ ꯑꯦꯚꯔꯦꯖꯗ ꯀꯌꯥ ꯆꯪꯒꯅꯤ?
ꯄꯥꯎꯈꯨꯝ ꯎꯠꯄ
ꯄꯥꯎꯈꯨꯝ k ꯁꯨꯕ ꯕꯦꯠꯇ ꯃꯥꯏꯄꯥꯛꯂꯕꯗꯤ ꯌꯨꯅꯤꯠ k ꯀꯥꯟꯅꯩ: +1, +2, +3 ꯑꯁꯤꯅꯆꯤꯡꯕ꯫ $10 ꯑꯃꯁꯨꯡ $500 ꯗ ꯁꯥꯏꯀꯜ ꯑꯁꯤ ꯃꯥꯏꯊꯤꯕ ꯃꯉꯥꯒꯤ ꯃꯇꯨꯡꯗ ꯁꯣꯛꯏ, ꯁꯥꯏꯀꯂꯒꯤ 3.57%, −$570 ꯒꯤ꯫ ꯑꯦꯚꯔꯦꯖꯗ ꯁꯥꯏꯀꯜ ꯑꯃꯅ ꯊꯝꯂꯕ $85.73 ꯒꯤ ꯃꯅꯨꯡꯗꯒꯤ $2.32 ꯃꯥꯡꯏ: 2.70%꯫
k ꯁꯨꯕ ꯊꯝꯂꯕ ꯁꯦꯜ ꯑꯁꯤ ꯌꯨꯅꯤꯠ 2ᵏ − 1: 1, 3, 7, 15, 31꯫ ꯑꯍꯥꯅꯕ k − 1 ꯊꯝꯂꯕ ꯁꯦꯜ ꯄꯨꯟꯁꯤꯜꯂꯕꯗ 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1꯫ k ꯁꯨꯕ ꯕꯦꯠꯇ ꯃꯥꯏꯄꯥꯛꯂꯕꯗꯤ 2ᵏ − 1 ꯄꯤ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯁꯥꯏꯀꯜ ꯑꯁꯤ (2ᵏ − 1) − (2ᵏ − k − 1) = ꯌꯨꯅꯤꯠ k ꯀꯥꯟꯅꯗꯨꯅ ꯂꯣꯏ: ꯃꯥꯏꯄꯥꯛꯄꯒꯤ ꯃꯃꯥꯡꯒꯤ ꯃꯥꯏꯊꯤꯕ ꯈꯨꯗꯤꯡꯒꯤꯗꯃꯛ ꯍꯦꯟꯅꯥ ꯌꯨꯅꯤꯠ ꯑꯃꯥ꯫
ꯌꯨꯅꯤꯠ ꯑꯃꯗ $10 ꯗ ꯊꯝꯂꯕ ꯁꯦꯜ ꯑꯁꯤ $10, $30, $70, $150 ꯑꯃꯁꯨꯡ $310; ꯃꯊꯪꯒꯤ, $630, ꯑꯁꯤ $500 ꯒꯤ ꯈ꯭ꯕꯥꯏꯗꯒꯤ ꯋꯥꯡꯕꯗꯒꯤ ꯍꯦꯟꯏ꯫ ꯃꯈꯥ ꯆꯠꯅꯅ ꯃꯥꯏꯊꯤꯕ ꯃꯉꯥ, (19/37)⁵ = 3.57%, ꯅ 10 + 30 + 70 + 150 + 310 = $570 ꯆꯪꯏ꯫
ꯑꯦꯚꯔꯦꯖ: k = 1 ꯗꯒꯤ 5 ꯒꯤꯗꯃꯛ ꯃꯥꯏꯄꯥꯛꯄꯁꯤꯡꯅ (19/37)ᵏ⁻¹ × 18/37 ꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯒꯥ ꯂꯣꯌꯅꯅ $10 × k ꯄꯨꯔꯀꯏ, ꯑꯃꯁꯨꯡ ꯁꯣꯀꯄꯅ 3.57% ꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯒꯥ ꯂꯣꯌꯅꯅ $570 ꯂꯧꯏ; ꯄꯨꯟꯅ −$2.32꯫ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ ꯊꯝꯂꯕ ꯁꯦꯜꯒꯤ ꯑꯦꯚꯔꯦꯖ $85.73, ꯑꯃꯁꯨꯡ 2.32/85.73 = 2.70%꯫ ꯇꯦꯕꯜ ꯑꯗꯨꯃꯛꯇ ꯃꯇꯝ ꯄꯨꯝꯕꯒꯤ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ ꯑꯁꯤ ꯃꯥꯏꯊꯤꯕ ꯇꯔꯨꯛꯀꯤ ꯃꯇꯨꯡꯗ ꯁꯣꯛꯏ, ꯁꯥꯏꯀꯂꯒꯤ 1.83%, −$630 ꯒꯤ, ꯑꯃꯁꯨꯡ ꯊꯝꯂꯕ $64.20 ꯒꯤ ꯃꯅꯨꯡꯗꯒꯤ $1.74 ꯃꯥꯡꯏ: ꯑꯃꯨꯛ ꯍꯟꯅꯥ 2.70%꯫
ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯉꯝꯈꯩ $1,000 ꯂꯩꯕ ꯇꯦꯕꯜ ꯑꯃꯗ ꯑꯍꯥꯅꯕ ꯕꯦꯠ $1 ꯒꯥ ꯂꯣꯌꯅꯅ ꯑꯉꯥꯡꯕꯗ ꯃꯥꯔꯇꯤꯟꯒꯦꯜ ꯑꯃꯥ ꯊꯧꯔꯥꯡ ꯇꯧꯏ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯕꯦꯠ ꯇꯔꯥ ꯌꯥꯏ: $1, $2, $4 ꯑꯁꯤꯅꯆꯤꯡꯕ $512 ꯐꯥꯑꯣꯕ꯫ ꯁꯥꯏꯀꯜ ꯑꯃꯅ ꯇꯔꯥ ꯄꯨꯝꯅꯃꯛ ꯀꯌꯥ ꯌꯥꯃꯅ ꯃꯥꯏꯊꯤꯕꯒꯅꯤ, ꯃꯁꯤꯗ ꯑꯦꯚꯔꯦꯖꯗ ꯀꯌꯥ ꯁꯥꯡꯅ ꯆꯪꯒꯅꯤ, ꯑꯃꯁꯨꯡ ꯕꯦꯠ 200 ꯒꯤ ꯅꯨꯡꯊꯤꯜ ꯑꯃꯥ ꯀꯌꯥ ꯌꯥꯃꯅ ꯃꯑꯣꯡ ꯑꯁꯤꯗ ꯂꯣꯏꯒꯅꯤ?
ꯄꯥꯎꯈꯨꯝ ꯎꯠꯄ
ꯄꯥꯎꯈꯨꯝ ꯁꯥꯏꯀꯜ ꯑꯃꯅ ꯇꯔꯥ ꯄꯨꯝꯅꯃꯛ 784 ꯗ ꯑꯃꯨꯛ ꯈꯔ ꯃꯥꯏꯊꯤꯕꯏ, ꯃꯁꯤꯅ $1,023 ꯆꯪꯏ꯫ ꯃꯁꯤ ꯑꯦꯚꯔꯦꯖꯗ ꯕꯦꯠ 1,600 ꯈꯔꯅꯤ, ꯑꯃꯁꯨꯡ ꯕꯦꯠ 200 ꯒꯤ ꯅꯨꯡꯊꯤꯜ ꯑꯃꯅ ꯃꯇꯝ 11% ꯈꯔꯗ ꯉꯝꯈꯩ ꯑꯗꯨ ꯊꯨꯡꯏ꯫ ꯁꯥꯏꯀꯜ ꯑꯃꯅ ꯍꯧꯖꯤꯀꯁꯨ ꯑꯦꯚꯔꯦꯖꯗ ꯊꯝꯂꯕ $11.31 ꯒꯤ ꯃꯅꯨꯡꯗꯒꯤ $0.31 ꯃꯥꯡꯏ: 2.70%꯫
ꯑꯉꯥꯡꯕꯗ ꯊꯝꯕ ꯕꯦꯠ ꯑꯃꯅ ꯖꯤꯔꯣ ꯌꯥꯎꯅ 19/37 ꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯒꯥ ꯂꯣꯌꯅꯅ ꯃꯥꯏꯊꯤꯕꯏ꯫ ꯃꯈꯥ ꯆꯠꯅꯅ ꯇꯔꯥ: (19/37)¹⁰ ≈ 0.00128, 784 ꯗ 1 ꯈꯔ꯫ ꯊꯝꯂꯕ ꯁꯦꯜ 1 + 2 + 4 + … + 512 ꯄꯨꯟꯁꯤꯜꯂꯕꯗ $1,023꯫
ꯁꯥꯏꯀꯜ ꯑꯃꯅ ꯑꯦꯚꯔꯦꯖꯗ ꯕꯦꯠ 2.05 ꯆꯪꯏ, ꯃꯔꯝ ꯑꯗꯨꯅ ꯁꯥꯏꯀꯜ 784 ꯅ ꯕꯦꯠ 1,600 ꯈꯔ ꯆꯪꯏ꯫ ꯁ꯭ꯄꯤꯟ ꯑꯃꯃꯛ ꯃꯁꯤꯡ ꯊꯤꯔꯕꯗ, ꯕꯦꯠ 200 ꯗ ꯀꯔꯤꯒꯨꯝꯕ ꯃꯐꯝ ꯑꯃꯗ ꯃꯥꯏꯊꯤꯕ ꯇꯔꯥꯒꯤ ꯁ꯭ꯇ꯭ꯔꯤꯛ ꯑꯃꯥ ꯃꯇꯝ 11.3% ꯗ ꯊꯣꯛꯏ꯫
ꯑꯋꯥꯕ ꯑꯆꯧꯕꯁꯤꯡꯒꯤ ꯃꯔꯛꯇ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯗꯨꯅ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ $1 ꯃꯥꯏꯄꯥꯀꯏ: ꯐꯠꯇꯕ ꯑꯗꯨꯒꯤ ꯃꯃꯥꯡꯗ $783 ꯈꯔ, ꯃꯇꯨꯡꯗ −$1,023, ꯃꯇꯝ ꯄꯨꯝꯅꯃꯛꯇ −$240 ꯈꯔ꯫ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31꯫ ꯁꯥꯏꯀꯜ ꯈꯨꯗꯤꯡꯗ ꯊꯝꯂꯕ ꯁꯦꯜ: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31꯫ ꯑꯃꯁꯨꯡ −0.31/11.31 = −2.70%꯫
$500 ꯒꯤ ꯉꯝꯈꯩꯗ ꯑꯍꯥꯅꯕ ꯕꯦꯠ $10, ꯕꯦꯠ ꯇꯔꯨꯛꯀꯥ ꯆꯥꯡꯗꯝꯕꯤꯌꯨ: ꯉꯝꯈꯩ ꯑꯗꯨ ꯁꯥꯏꯀꯜ 55 ꯗ ꯑꯃꯨꯛ ꯂꯥꯀꯏ, ꯑꯃꯁꯨꯡ ꯕꯦꯠ 200 ꯒꯤ ꯅꯨꯡꯊꯤꯜ ꯑꯃꯅ ꯃꯇꯝ 84% ꯗ ꯃꯁꯤ ꯊꯨꯡꯏ꯫ ꯍꯦꯟꯅ ꯋꯥꯡꯕ ꯉꯝꯈꯩ ꯑꯃꯅ ꯁꯣꯀꯄ ꯑꯗꯨ ꯍꯦꯟꯅ ꯇꯞꯅ ꯑꯃꯁꯨꯡ ꯍꯦꯟꯅ ꯆꯥꯎꯅ ꯇꯧꯏ, ꯃꯃꯜ ꯍꯟꯊꯍꯜꯂꯗꯦ꯫

ꯅꯣꯛꯅꯕꯒꯤ ꯀꯃꯤꯛ

ꯀꯔꯤꯒꯨꯝꯕ ꯉꯝꯈꯩ ꯑꯃꯗꯁꯨ ꯍꯦꯟꯅ ꯈꯟꯅꯕ ꯑꯃꯁꯨꯡ ꯍꯦꯟꯅ ꯆꯥꯎꯕ꯫ ꯃꯜꯇꯤꯞꯂꯥꯏ ꯑꯗꯨ ꯍꯣꯡꯗꯦ꯫

ꯍꯧꯔꯛꯐꯃꯁꯤꯡ

Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010꯫ ꯆꯦꯞꯇꯔ 13, ꯔꯨꯂꯦꯠ꯫
Thorp, E. O. “Optimal gambling systems for favorable games.” Review of the International Statistical Institute 37(3), 1969.
Thorp, E. O. The Mathematics of Gambling. Gambling Times / Lyle Stuart, 1984.
Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013.

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