ꯁꯃꯁ꯭ꯌꯥ: ꯖꯤꯔꯣ ꯊꯣꯛꯅꯕ ꯀꯌꯥ ꯉꯥꯏꯒꯗꯒꯦ
ꯑꯦꯚꯔꯦꯖꯗ ꯖꯤꯔꯣ ꯑꯁꯤ ꯁ꯭ꯄꯤꯟ 37 ꯗ ꯑꯃꯨꯛ ꯊꯣꯀꯏ꯫ ꯑꯁꯦꯡꯕꯗ ꯀꯌꯥ ꯉꯥꯏꯒꯗꯒꯦ, ꯑꯃꯁꯨꯡ ꯃꯁꯤꯅ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ ꯊꯝꯕ ꯃꯤ ꯑꯃꯗ ꯀꯔꯤ ꯇꯧꯕꯒꯦ?
ꯖꯤꯔꯣꯒꯤꯗꯃꯛ ꯉꯥꯏꯕ ꯑꯁꯤ ꯖꯤꯑꯣꯃꯦꯠꯔꯤꯛ ꯑꯣꯏ: ꯑꯦꯚꯔꯦꯖ ꯁ꯭ꯄꯤꯟ 37, ꯃꯤꯗꯤꯌꯥꯟ 25, ꯑꯃꯁꯨꯡ ꯉꯥꯏꯕ ꯀꯨꯟꯗ ꯑꯃꯥ ꯁ꯭ꯄꯤꯟ 110 ꯗꯒꯤ ꯍꯦꯟꯏ꯫ ꯁ꯭ꯄꯤꯟ 200 ꯒꯤ ꯁꯦꯁꯟ ꯑꯃꯗ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ ꯊꯝꯕ ꯃꯤ ꯑꯃꯅ ꯖꯤꯔꯣ 5.4 ꯊꯥꯖꯒꯅꯤ ꯑꯃꯁꯨꯡ 1 ꯗꯒꯤ 10 ꯐꯥꯑꯣꯕꯒꯤ ꯃꯅꯨꯡꯗ ꯀꯔꯤꯒꯨꯝꯕ ꯑꯃꯥ ꯎꯒꯅꯤ꯫
ꯁꯃꯁ꯭ꯌꯥ
ꯖꯤꯔꯣꯒꯤꯆꯛꯔꯖꯤꯔꯣꯆꯞ ꯃꯥꯟꯅꯕ, ꯗꯖꯟ ꯅꯠꯔꯒ ꯀꯂꯝ ꯕꯦꯠ ꯑꯃꯠꯇꯅ ꯀꯨꯞꯇꯕ ꯑꯁꯪꯕ ꯄꯣꯀꯦꯠ ꯑꯃꯈꯛꯇ; ꯍꯥꯎꯁ ꯑꯦꯖꯀꯤ ꯍꯧꯔꯀꯐꯝꯅꯤ꯫ ꯑꯇꯣꯞꯄ ꯃꯃꯤꯡ: ꯁꯤꯡꯒꯜ ꯖꯤꯔꯣ, 0꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯈꯨꯗꯣꯡꯆꯥꯕꯃꯦꯊꯃꯦꯇꯤꯛꯁꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕꯃꯍꯩ ꯑꯃꯥ ꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ, 0 ꯗꯒꯤ 1 ꯐꯥꯑꯣꯕ ꯅꯠꯔꯒ 0% ꯗꯒꯤ 100% ꯐꯥꯑꯣꯕ꯫ ꯃꯁꯤꯡ ꯑꯃꯥ: 1/37 = 2.70%꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯁꯤ ꯁ꯭ꯄꯤꯟ ꯈꯨꯗꯤꯡꯗ 1/37 ꯑꯣꯏ꯫ (a) ꯃꯊꯪꯒꯤ ꯁ꯭ꯄꯤꯟ 37 ꯗ ꯖꯤꯔꯣ ꯑꯃꯠꯇ ꯊꯣꯛꯇꯕꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ ꯀꯌꯥꯅꯣ? (b) ꯃꯤꯗꯤꯌꯥꯟ ꯉꯥꯏꯕ ꯀꯌꯥꯅꯣ? (c) ꯁ꯭ꯄꯤꯟ 200 ꯒꯤ ꯆꯞ ꯃꯥꯟꯅꯕ ꯁꯦꯁꯟ ꯑꯃꯗ ꯁꯥꯟꯅꯔꯤꯕ ꯑꯃꯅ ꯖꯤꯔꯣ ꯀꯌꯥ ꯊꯥꯖꯒꯗꯒꯦ, ꯑꯃꯁꯨꯡ ꯔꯦꯟꯖ ꯑꯗꯨ ꯀꯌꯥꯅꯣ?
ꯁꯣꯜꯌꯨꯁꯅꯁꯤꯡ
(a) (36/37)³⁷ = 36.3%꯫ ꯃꯇꯃꯒꯤ ꯑꯍꯨꯃꯗ ꯑꯃꯗꯒꯤ ꯍꯦꯟꯅ, ꯁ꯭ꯄꯤꯟ 37 ꯑꯁꯤ ꯖꯤꯔꯣ ꯑꯃꯠꯇ ꯊꯣꯛꯇꯅ ꯆꯠꯈꯤ, ꯃꯁꯤ “ꯁ꯭ꯄꯤꯟ 37 ꯗ ꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯕꯦꯠꯁ꯭ꯇ꯭ꯔꯦꯠ-ꯑꯞꯃꯁꯤꯡ ꯑꯃꯈꯛꯇ, 35:1 ꯄꯦꯑꯥꯎꯠ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯗ ꯌꯥꯃꯗ꯭ꯔꯕꯗ ꯑꯃꯨꯛ ꯊꯣꯛꯄ” ꯑꯁꯤ 63.7% ꯈꯛꯇꯃꯛ ꯑꯣꯏꯕꯒꯥ ꯃꯁꯤꯡ ꯊꯤꯕ ꯑꯗꯨꯃꯛꯅꯤ꯫ (b) ꯉꯥꯏꯕ W ꯑꯁꯤ ꯖꯤꯑꯣꯃꯦꯠꯔꯤꯛ ꯑꯣꯏ; p(W > k) = (36/37)ᵏ; ꯃꯤꯗꯤꯌꯥꯟ ꯑꯁꯤ ꯃꯁꯤꯅ 0.5 ꯂꯥꯟꯊꯣꯀꯄ k ꯑꯗꯨꯅꯤ, ꯍꯥꯌꯕꯗꯤ k ≈ 25꯫ ꯉꯥꯏꯕꯁꯤꯡꯒꯤ ꯇꯉꯥꯏ ꯁ꯭ꯄꯤꯟ 25 ꯅꯠꯔꯒ ꯃꯁꯤꯗꯒꯤ ꯊꯨꯕ ꯑꯃꯁꯨꯡ ꯇꯉꯥꯏ ꯍꯦꯟꯅ ꯁꯥꯡꯏ; 37 ꯒꯤ ꯑꯦꯚꯔꯦꯖ ꯑꯁꯤ ꯁꯥꯡꯕ ꯃꯃꯥꯏꯅ ꯋꯥꯡꯍꯜꯂꯤ꯫ (c) ꯁ꯭ꯄꯤꯟ 200 ꯗ ꯖꯤꯔꯣꯁꯤꯡ ꯑꯁꯤ ꯕꯥꯏꯅꯣꯃꯤꯑꯦꯜ ꯑꯣꯏ, ꯑꯦꯚꯔꯦꯖ 200/37 = 5.4 ꯑꯃꯁꯨꯡ ꯁ꯭ꯇꯦꯟꯗꯔꯗ ꯗꯤꯚꯤꯌꯦꯁꯟꯃꯦꯊꯃꯦꯇꯤꯛꯁꯁ꯭ꯇꯦꯟꯗꯔꯗ ꯗꯤꯚꯤꯌꯦꯁꯟꯚꯦꯌꯔꯤꯌꯦꯟꯁꯀꯤ ꯁ꯭ꯀ꯭ꯋꯦꯌꯔ ꯔꯨꯠ; ꯃꯍꯩ ꯑꯃꯅ ꯐꯪꯒꯗꯕ ꯃꯃꯜꯗꯒꯤ ꯌꯥꯝꯅ ꯂꯥꯞꯄ ꯃꯑꯣꯡ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → √(200 × (1/37) × (36/37)) ≈ 2.3꯫ 95% ꯒꯤ ꯔꯦꯟꯖ ꯑꯁꯤ ꯃꯔꯥꯡ ꯀꯥꯏꯅ 1 ꯗꯒꯤ 10 ꯐꯥꯑꯣꯕꯅꯤ꯫
ꯖꯤꯔꯣꯅ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ ꯊꯝꯕ ꯃꯤ ꯑꯃꯗ ꯀꯌꯥ ꯃꯥꯡꯍꯜꯂꯤꯕꯒꯦ
ꯖꯤꯔꯣ ꯈꯨꯗꯤꯡꯃꯛꯅ ꯊꯝꯂꯕ ꯁꯦꯜ ꯑꯃꯥ ꯃꯥꯡꯍꯜꯂꯤ, ꯑꯃꯁꯨꯡ ꯑꯉꯥꯡꯕꯗꯕꯦꯠꯑꯉꯥꯡꯕꯑꯉꯥꯡꯕ ꯃꯁꯤꯡ 18 ꯗ ꯊꯝꯕ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠ꯫ ꯐ꯭ꯔꯦꯟꯆ: ꯔꯨꯖ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯍꯥꯎꯁ ꯑꯦꯖ ꯄꯨꯝꯅꯃꯛ ꯑꯁꯤ ꯖꯤꯔꯣꯁꯤꯡꯅꯤ: ꯑꯦꯚꯔꯦꯖ ꯁ꯭ꯄꯤꯟ 37 ꯒꯤ ꯃꯅꯨꯡꯗ ꯃꯥꯏꯄꯥꯛꯄ 18 ꯑꯃꯁꯨꯡ ꯃꯥꯏꯊꯤꯕ 18 ꯅ ꯑꯃꯅ ꯑꯃꯕꯨ ꯃꯨꯠꯊꯠꯂꯤ, ꯑꯃꯁꯨꯡ ꯖꯤꯔꯣ ꯑꯃꯗꯤ −1/37 ꯑꯗꯨꯅꯤ꯫ $10 ꯒꯤ ꯁ꯭ꯄꯤꯟ 200 ꯒꯤ ꯁꯦꯁꯟ ꯑꯃꯗ ꯊꯥꯖꯒꯗꯕ ꯖꯤꯔꯣ 5.4 ꯅ $54 ꯃꯥꯡꯍꯜꯂꯤ, ꯃꯁꯤ ꯆꯨꯝꯅ 200 × $10 / 37 ꯅꯤ꯫ ꯖꯤꯔꯣ 2 ꯂꯩꯕ ꯁꯦꯁꯟ ꯑꯃꯅ $34 ꯒꯤ ꯂꯥꯛ ꯐꯕ ꯐꯪꯈꯤ; ꯖꯤꯔꯣ 9 ꯂꯩꯕ ꯁꯦꯁꯟ ꯑꯃꯅ $36 ꯒꯤ ꯂꯥꯛ ꯐꯠꯇꯕ ꯐꯪꯈꯤ, ꯑꯃꯁꯨꯡ ꯃꯁꯤ ꯃꯤꯅꯝꯕ ꯂꯩꯕꯒꯨꯝ ꯐꯥꯑꯣꯒꯅꯤ꯫ ꯂꯥ ꯄꯥꯔꯇꯥꯖꯅꯤꯌꯝꯂꯥ ꯄꯥꯔꯇꯥꯖꯐ꯭ꯔꯦꯟꯆ ꯅꯤꯌꯝ: ꯖꯤꯔꯣ ꯊꯣꯛꯂꯕꯗꯤ ꯆꯞ ꯃꯥꯟꯅꯕ ꯕꯦꯠꯀꯤ ꯊꯝꯂꯕ ꯁꯦꯜꯒꯤ ꯁꯔꯨꯛ ꯑꯍꯦꯟꯕ ꯍꯜꯂꯀꯏ꯫ ꯕꯦꯠ ꯑꯗꯨꯁꯤꯡꯒꯤ ꯍꯥꯎꯁ ꯑꯦꯖ 1.35% ꯐꯥꯑꯣꯕ ꯁꯔꯨꯛ ꯑꯅꯤꯗ ꯊꯥꯖꯤꯜꯂꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯇꯦꯕꯜ ꯑꯃꯗ ꯖꯤꯔꯣ ꯈꯨꯗꯤꯡꯃꯛꯅ ꯊꯝꯂꯕ ꯁꯦꯜꯒꯤ ꯇꯉꯥꯏ ꯃꯥꯡꯍꯜꯂꯤ ꯑꯃꯁꯨꯡ ꯃꯁꯤꯡꯁꯤꯡ ꯑꯗꯨ ꯇꯉꯥꯏ ꯑꯣꯏꯔꯛꯏ꯫
ꯃꯇꯨꯡꯒꯤ ꯁꯃꯁ꯭ꯌꯥ
ꯃꯇꯨꯡ ꯏꯟꯅ ꯖꯤꯔꯣ ꯑꯅꯤ ꯊꯣꯛꯄꯒꯤ ꯈꯨꯗꯣꯡꯆꯥꯕ ꯀꯌꯥꯅꯣ? (1/37)² = 1/1,369꯫ ꯄꯨꯡ ꯑꯃꯗ ꯁ꯭ꯄꯤꯟ 50 ꯆꯠꯂꯤꯕ ꯇꯦꯕꯜ 20 ꯂꯩꯕ ꯀꯦꯁꯤꯅꯣ ꯑꯃꯗ, ꯃꯁꯤ ꯄꯨꯡ 1.4 ꯈꯨꯗꯤꯡꯗ ꯀꯗꯥꯏꯗꯥ ꯃꯇꯨꯡ ꯏꯟꯅ ꯖꯤꯔꯣ ꯑꯅꯤꯒꯤ ꯄꯦꯌꯔ ꯑꯃꯥ ꯃꯔꯥꯡ ꯀꯥꯏꯅꯤ꯫ ꯃꯁꯤ ꯎꯔꯤꯕ ꯁꯥꯟꯅꯔꯤꯕꯁꯤꯡꯅ ꯆꯛꯔ ꯑꯗꯨ ꯁꯣꯛꯂꯦ ꯍꯥꯌꯅ ꯂꯧꯏ; ꯃꯁꯤꯡ ꯊꯤꯕꯅ ꯍꯥꯌꯔꯤ ꯃꯗꯨꯗꯤ ꯉꯁꯤ ꯂꯩꯄꯥꯛꯄꯣꯛꯄ ꯈꯛꯇꯅꯤ꯫
ꯇꯧꯔꯒ ꯌꯦꯡꯕꯤꯌꯨ
ꯁꯦꯁꯟ ꯁꯤꯃꯨꯂꯦꯇꯔꯅ ꯁꯦꯁꯟ ꯈꯨꯗꯤꯡꯗ ꯖꯤꯔꯣ ꯃꯁꯤꯡ ꯊꯤꯔꯤ꯫ ꯁ꯭ꯄꯤꯟ 100 ꯑꯁꯤ ꯑꯃꯨꯛ ꯂꯤꯁꯤꯡ ꯈꯔꯥ ꯆꯂꯥꯏꯗꯨꯅ ꯃꯁꯤꯡ ꯑꯗꯨ ꯄ꯭ꯂꯣꯠ ꯇꯧꯕꯤꯌꯨ: ꯃꯊꯛꯇ ꯂꯩꯕ ꯍꯤꯁ꯭ꯇꯣꯒ꯭ꯔꯥꯝ ꯑꯗꯨ, ꯃꯁꯤꯒꯤ ꯑꯔꯣꯏꯕ ꯃꯈꯥꯗ ꯁꯥꯡꯕ ꯃꯃꯥꯏꯒꯥ ꯂꯣꯌꯅꯅ ꯊꯣꯔꯛꯀꯅꯤ꯫ ꯋꯥꯔꯤꯁꯤꯡ ꯂꯥꯀꯄ ꯃꯐꯝ ꯑꯗꯨ ꯃꯃꯥꯏ ꯑꯗꯨꯅꯤ꯫
ꯐꯣꯔꯃꯨꯂꯥ
ꯋꯥꯍꯪꯁꯤꯡ
ꯆꯨꯝꯃꯤꯟꯅꯕ ꯑꯃꯗꯗꯤ ꯅꯠꯇꯦ꯫ ꯕꯥꯏꯌꯥꯁ ꯂꯩꯕ ꯆꯛꯔꯂꯦꯕꯕꯥꯏꯌꯥꯁ ꯂꯩꯕ ꯆꯛꯔꯁꯛꯇꯝꯒꯤ ꯑꯁꯣꯏꯕꯁꯤꯡꯅ ꯄꯣꯀꯦꯠ ꯈꯔꯕꯨ 1/37 ꯗꯒꯤ ꯍꯦꯟꯅꯥ ꯊꯣꯛꯍꯟꯕ ꯆꯛꯔ꯫ ꯌꯥꯝꯅ ꯌꯥꯝꯗꯦ, ꯊꯝꯂꯕ ꯁ꯭ꯄꯤꯟ ꯂꯤꯁꯤꯡ ꯀꯌꯥꯅ ꯈꯛꯇꯃꯛ ꯈꯉꯕ ꯉꯝꯃꯤ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯗ ꯀꯔꯤꯒꯨꯝꯕ ꯄꯣꯀꯦꯠꯈꯨꯠꯂꯥꯏꯄꯣꯀꯦꯠꯔꯣꯇꯔꯗ ꯂꯩꯕ ꯃꯁꯤꯡ ꯏꯔꯕ ꯀꯣꯠ 37 ꯒꯤ ꯃꯅꯨꯡꯗ ꯑꯃꯥ꯫ ꯑꯇꯣꯞꯄ ꯃꯃꯤꯡ: ꯁ꯭ꯂꯣꯠ, ꯀꯞ꯫ꯃꯄꯨꯡ ꯄꯥꯕꯤꯌꯨ → ꯑꯃꯃꯛ ꯑꯣꯏꯕ ꯌꯥꯏ, ꯑꯃꯁꯨꯡ ꯃꯁꯤ ꯈꯉꯅꯕ ꯁ꯭ꯄꯤꯟ ꯂꯤꯁꯤꯡ ꯀꯌꯥ ꯃꯊꯧ ꯇꯥꯒꯅꯤ꯫
ꯃꯁꯤ ꯑꯇꯩ ꯄꯨꯝꯅꯃꯛꯒꯨꯝꯅ 2.70% ꯒꯤ ꯕꯦꯠ ꯑꯃꯅꯤ꯫ ꯃꯁꯤꯅ ꯁꯦꯁꯟꯒꯤ ꯃꯑꯣꯡ ꯍꯣꯡꯍꯜꯂꯤ (ꯖꯤꯔꯣ ꯑꯃꯥ ꯃꯥꯏꯄꯥꯛꯄ ꯑꯣꯏꯔꯛꯏ), ꯃꯃꯜ ꯑꯗꯨ ꯅꯠꯇꯦ꯫
ꯃꯔꯃꯗꯤ ꯉꯥꯏꯕꯁꯤꯡꯒꯤ ꯗꯤꯁ꯭ꯇ꯭ꯔꯤꯕ꯭ꯌꯨꯁꯟ ꯑꯁꯤ ꯁꯦꯡꯅ ꯃꯥꯟꯅꯗꯦ: ꯊꯨꯕ ꯉꯥꯏꯕ ꯀꯌꯥ, ꯌꯥꯝꯅꯥ ꯁꯥꯡꯕ ꯈꯔꯥ꯫
ꯅꯣꯛꯅꯕꯒꯤ ꯀꯃꯤꯛ
ꯑꯦꯚꯔꯦꯖ 37, ꯃꯤꯗꯤꯌꯥꯟ 25, ꯑꯃꯁꯨꯡ ꯋꯥꯔꯤꯁꯤꯡ ꯍꯤꯡꯂꯤꯕ ꯁꯥꯡꯕ ꯃꯃꯥꯏ ꯑꯃꯥ꯫
ꯍꯧꯔꯛꯐꯃꯁꯤꯡ
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.