ᕉᓚᑦ: ᓈᓴᐃᓂᖅ · ᓈᓴᐃᓂᕐᒧᑦ ᐊᐱᖅᑯᑎ · 10 ᒥᓂᑦ

ᐊᐱᖅᑯᑎ: ᐱᙳᐊᕐᕕᐅᑉ ᑭᒡᓕᖓ

$10-ᒥᒃ ᒫᑎᓐᒐᐃᓪ $500 ᐱᙳᐊᕐᕕᒻᒥ. ᖃᓄᖅ ᐊᑲᐅᓪᓗ ᓱᕋᒃᐸᓚᐅᕐᒪᖔᑦ, ᓱᕋᒃᑎᓪᓗᒍ ᖃᓄᐃᓪᓗ ᐊᒥᓱᓂᒃ ᐊᓯᐅᔾᔨᓂᐊᕐᒪᖔᑦ, ᑭᒡᓕᖓᓗ ᖁᑦᑎᓕᖅᑎᑕᐅᒃᐸᑦ ᐃᑲᔪᕐᓂᐊᕐᒪᖔᑦ?

ᓇᐃᑦᑐᖅ ᑭᐅᔾᔪᑎ

ᐱᖓᓲᔪᖅᑐᑦ ᐊᓯᐅᔾᔨᓃᑦ ᑐᓪᓕᖅᑐᑦ ᓱᕋᒃᑎᑦᓯᔪᑦ: ᐱᙳᐊᕈᓯᕐᒧᑦ 1.8%, ᖃᓂᒋᔭᖓ 100 ᑲᐃᕙᕐᓂᓕᓐᓂ ᐱᙳᐊᖅᑕᐅᓂᖏᓐᓂ 60%. ᓱᕋᒃᑐᖅ $630-ᒥᒃ ᐊᑭᖃᖅᑐᖅ. ᑭᒡᓕᖓ $5,000-ᒧᑦ ᖁᑦᑎᓕᖅᑎᑕᐅᒃᐸᑦ ᓱᕋᒃᑐᖅ ᐃᒪᐃᓐᓂᖅᓴᐅᓕᖅᑐᖅ (ᐱᙳᐊᕈᓯᕐᒧᑦ 0.25%) ᐊᖏᓂᖅᓴᐅᓕᖅᑐᓗ ($5,110). ᐱᙳᐊᕈᑎᓕᐊᕆᔭᐅᔪᒧᑦ ᑕᐃᓚᒧᑦ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᖅ 2.70% ᖃᓄᐃᑦᑐᒥ ᑭᒡᓕᖓᓂ.

ᐱᒋᐊᕐᕕᒃ / ᑭᒡᓕᖓ

$10 / $500

ᓱᕋᒃᑎᑦᓯᔪᑦ ᐊᓯᐅᔾᔨᓃᑦ

ᐱᙳᐊᕈᓯᕐᒧᑦ p(ᓱᕋᒃᑐᖅ)

1.8%

ᓱᕋᒃᑎᓪᓗᒍ ᐊᓯᐅᔾᔨᓂᖅ

$630

ᐊᐱᖅᑯᑎ

ᐱᙳᐊᖅᑎ $10-ᒥᒃ ᐊᐅᐸᓗᒃᑐᒧᑦᐱᙳᐊᕈᑎᐊᐅᐸᓗᒃᑐᖅᐊᔾᔨᒌᒃᑐᒥᒃ ᐊᑭᓕᐅᑎᓕᒃ ᐱᙳᐊᕈᑎ 18 ᐊᐅᐸᓗᒃᑐᓄᑦ ᓈᓴᐅᑎᓄᑦ. ᕖᓇᓯᑐᑦ: ᕈᔾ (rouge).ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᐱᙳᐊᕈᑎᓕᐅᖅᑐᖅ, ᐊᓯᐅᔾᔨᑐᐊᕐᓗᓂ ᒪᕐᕉᕆᐊᖅᓯᓪᓗᓂ, ᐊᔪᒐᑐᐊᕐᓗᓂᓗ ᐱᒋᐊᖅᑭᓪᓗᓂ. ᐱᙳᐊᕐᕕᐅᑉ ᐊᖏᓛᖅ ᑭᒡᓕᖓ $500. (a) ᐊᓯᐅᔾᔨᓃᑦ ᖃᑦᓯᐅᔪᑦ ᑐᓪᓕᖅᑐᑦ ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅᐸᕐᓇᐅᑎᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅᒪᓕᒐᖅ ᓇᓕᐊᑐᐃᓐᓇᖅ ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖓᓂᒃ ᐊᓂᒍᖅᓯᒪᔪᑦ ᐱᓐᓂᖏᑦ ᒪᓕᒡᓗᒋᑦ ᐊᓯᑦᑐᐃᔪᖅ. ᐊᑲᐅᙱᑦᑐᑦ ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᐅᔪᑦ ᐊᓯᐅᔾᔨᓂᖅ ᑭᖑᓂᖓᓂ ᖁᕝᕙᑦᑎᔪᑦ, ᐱᐅᔪᑦ ᐊᔪᒐᕐᓂᖅ ᑭᖑᓂᖓᓂ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᓱᕋᒃᑎᑦᓯᕙᑦ? (b) ᐱᙳᐊᕈᓯᕐᒧᑦ ᐱᔪᓐᓇᕐᓂᖅᓈᓴᐃᓂᖅᐱᔪᓐᓇᕐᓂᖅᐱᓐᓂᐅᑉ ᐱᔪᓐᓇᕐᓂᖓ, 0-ᒥᑦ 1-ᒧᑦ ᐅᕝᕙᓘᓐᓃᑦ 0%-ᒥᑦ 100%-ᒧᑦ. ᓈᓴᐅᑎ ᐊᑕᐅᓯᖅ: 1/37 = 2.70%.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᖃᓄᐃᑦᑐᖅ? (c) ᓱᕋᒃᑎᓪᓗᒍ ᖃᓄᐃᓪᓗ ᐊᒥᓱᑦ ᐊᓯᐅᔾᔨᔭᐅᕙᑦ? (d) $5,000 ᑭᒡᓕᖓ ᑭᐅᔾᔪᑏᑦ ᖃᓄᖅ ᐊᓯᑦᑐᖅᓯᕙᑦ? (e) ᑭᒡᓕᖅ ᓇᓪᓕᐊᑐᐃᓐᓇᖅ ᐸᕐᓇᐅᑎ ᑮᓇᐅᔭᒃᓴᖅᑎᑦᓯᔪᒥᒃ ᐱᓕᖅᑎᑦᓯᕙ?

ᕉᓚᑦ: ᐱᙳᐊᕐᕕᐅᑉ ᑭᒡᓕᖏᑦ

ᑭᐅᔾᔪᑏᑦ (a)–(c)

ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑦ: 10, 20, 40, 80, 160, 320. ᐊᕐᕕᓂᓖᑦ ᐊᑕᐅᓯᖅ 640 > 500-ᖑᓂᐊᖅᑐᖅ, ᑕᐃᒪ ᐱᖓᓲᔪᖅᑐᑦ ᐊᓯᐅᔾᔨᓃᑦ ᓱᕋᒃᑎᑦᓯᔪᑦ. (b) p(ᐊᓯᐅᔾᔨᓂᖅ) = 19/37 ᐊᐅᐸᓗᒃᑐᒥ; (19/37)⁶ = 0.0183, ᖃᓂᒋᔭᖓ 1.8% ᐱᙳᐊᕈᓯᕐᒧᑦ. ᐱᙳᐊᕈᓯᖅ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᖃᓂᒋᔭᖓ ᒪᕐᕉᓂᒃ ᑲᐃᕙᕐᓂᓂᒃ ᐊᑯᓂᐅᑎᒋᔪᖅ, ᑕᐃᒪ 100 ᑲᐃᕙᕐᓂᓕᒃ ᐱᙳᐊᖅᑕᐅᓂᖓ ᖃᓂᒋᔭᖓ 50 ᐱᙳᐊᕈᓯᕐᓂᒃ ᐱᖃᖅᑐᖅ, p(ᐊᑕᐅᑦᑎᐊᕐᒥᒃ ᐅᖓᑖᓄᓪᓘᓐᓃᑦ ᓱᕋᒃᑐᖅ) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ᑕᒪᓐᓇ 63 ᐱᒋᐊᕐᕕᐅᔪᑦ ᐊᑦᔨᐅᑏᑦᑮᓇᐅᔭᖅᐊᑦᔨᐅᑎᐸᕐᓇᐅᑎᐅᑉ ᐅᕝᕙᓘᓐᓃᑦ ᐱᙳᐊᖅᑎᐅᑉ ᑐᙵᕕᒋᔭᖓ ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖓ. ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᐅᔪᑦ ᐊᑦᔨᐅᑎᓄᑦ ᓇᓗᓇᐃᖅᑕᐅᕙᒃᑐᑦ, ᑕᐃᒪ ᓇᓕᐊᓂᑐᐃᓐᓇᖅ ᐱᙳᐊᕐᕕᒻᒥ ᐊᑐᕈᓐᓇᖁᓪᓗᒋᑦ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ →, ᐱᙳᐊᕈᓯᖅ $10-ᑐᐊᒥᒃ ᐊᔪᒐᕐᓂᐊᕐᓂᕐᒥᒃ ᐱᓇᓱᒃᑎᓪᓗᒍ.

ᑭᒡᓕᖓᓱᕋᒃᑎᑦᓯᔪᑦ ᐊᓯᐅᔾᔨᓃᑦᐱᙳᐊᕈᓯᕐᒧᑦ p(ᓱᕋᒃᑐᖅ)ᓱᕋᒃᑎᓪᓗᒍ ᐊᓯᐅᔾᔨᓂᖅp(100 ᑲᐃᕙᕐᓂᓂ ᓱᕋᒃᑐᖅ)ᐅᑎᖅᑎᑦᓯᓂᕐᒧᑦ ᐊᔪᒐᕐᓃᑦ ᐱᔭᕆᐊᓕᑦ

$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

ᖁᓛ ᒪᕐᕉᕆᐊᖅᑎᑕᐅᑦᓯᐊᕐᒪᑦ ᓱᕋᒃᑐᖅ ᖃᓂᒋᔭᖓ ᓇᑉᐸᖓᓄᑦ ᐃᒪᐃᓐᓂᖅᓴᐅᓕᖅᑐᖅ ᒪᕐᕉᕆᐊᖅᓗᒍᓗ ᐊᑭᑦᑐᓂᖅᓴᐅᓕᖅᑐᖅ. ᐊᐱᕆᔭᐅᔪᖅ ᐊᓯᑦᑐᐊᓗᐊᙱᑦᑐᖅ.

ᑭᐅᔾᔪᑎ (d)

$5,000-ᒥ: ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑦ 2,560-ᒧᑦ ᐱᔪᓐᓇᖅᑐᑦ; ᖁᓕᑦ ᐱᙳᐊᕈᑎ, 5,120, ᐊᖏᖅᑕᐅᙱᑦᑐᖅ. ᐊᕐᕕᓂᓖᑦ ᐱᖓᓱᑦ ᐊᓯᐅᔾᔨᓃᑦ ᓱᕋᒃᑎᑦᓯᔪᑦ: (19/37)⁹ ≈ 0.25% ᐱᙳᐊᕈᓯᕐᒧᑦ, ᖃᓂᒋᔭᖓ 100 ᑲᐃᕙᕐᓂᓕᓐᓂ ᐱᙳᐊᖅᑕᐅᓂᖏᓐᓂ 11%. ᓱᕋᒃᑎᓪᓗᒍ ᐊᓯᐅᔾᔨᓂᖅ: 10 + 20 + … + 2,560 = $5,110. ᐱᙳᐊᖅᑎ ᐊᑦᑕᓇᖅᑐᒥᒃ ᐃᒪᐃᓐᓂᖅᓴᒥᒃ ᓂᐅᕕᖅᓯᒪᔪᖅ ᐊᖏᓂᖅᓴᐅᓂᖓᓂᒃ ᐊᑭᓕᓪᓗᓂ. ᐱᙳᐊᖅᑕᐅᓂᖏᓐᓂ ᐊᒥᓱᓂ ᓱᕋᒃᑐᑦ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᑭᖏᑦ p × ᐊᓯᐅᔾᔨᓂᖅ ≈ 0.0183 × 630 = $11.5 ᐱᙳᐊᕈᓯᕐᒧᑦ $500 ᑭᒡᓕᖓᓂ, 0.0025 × 5,110 = $12.8-ᓗ $5,000-ᒥ; ᒪᕐᕉᒃ ᖃᓂᒌᒃᑐᒃ ᑭᓯᐊᓂ ᐊᔾᔨᒌᙱᑦᑐᒃ, ᐊᑐᓂᓗ ᐱᙳᐊᕈᓯᐅᑉ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᖓᓈᓴᐃᓂᖅᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᖅᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᑭᖓ ᐊᕐᓕᒃᓗᒍ ᐱᙳᐊᕈᑏᑦ ᑲᑎᑕᐅᓂᖏᓐᓄᑦ; ᐱᙳᐊᖅᑕᐅᓂᖓᓂ ᐅᕝᕙᓘᓐᓃᑦ ᐱᕕᒃᓴᕐᒥ ᐊᓯᐅᔾᔨᓂᖅ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ →, ᐃᓚᓗᒍ $10 ᐊᔪᒐᖅᑐᑦ ᐱᙳᐊᕈᓰᑦ ᐱᔭᖏᑦ.

ᑭᐅᔾᔪᑎ (e)

ᐋᒃᑲ. ᐱᙳᐊᕈᓯᕐᒧᑦ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᖅ = (ᐱᙳᐊᕈᓯᕐᒥ ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑕ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐃᓘᓐᓇᖓ) / 37, ᖃᓄᐃᑦᑐᒥ ᑭᒡᓕᖓᓂ, ᐱᙳᐊᕈᑎ ᐊᑐᓂ ᓇᒻᒥᓂᖅ −1/37-ᖑᒻᒪᑦ. ᑭᒡᓕᖓ ᑐᐊᖅ ᓇᓚᐅᑦᑖᖅᑐᖅ ᐊᓯᐅᔾᔨᓂᖅ ᖃᓄᖅ ᐊᕕᒃᑕᐅᓂᐊᕐᒪᖔᑦ ᐊᒥᓱᓄᑦ ᒥᑭᔪᓄᑦ ᐊᔪᒐᕐᓂᕐᓄᑦ ᐊᒥᓲᙱᑦᑐᓄᓪᓗ ᐊᖏᔪᓄᑦ ᐊᓯᐅᔾᔨᓂᕐᓄᑦ. ᑭᒡᓕᖃᙱᑉᐸᑦ, ᐱᙳᐊᕈᑎᒃᓴᐃᓪᓗ ᐱᙳᐊᕈᑎᒃᓴᐃᑦᑮᓇᐅᔭᖅᐱᙳᐊᕈᑎᒃᓴᐃᑦᑮᓇᐅᔭᑦ ᐃᓘᓐᓇᖏᑦ ᐱᙳᐊᕐᓂᕐᒧᑦ ᐊᕕᒃᓯᒪᔭᐅᔪᑦ. ᐱᙳᐊᕈᑎᒃᓴᓂᒃ ᐊᐅᓚᑦᓯᓂᖅ ᑐᑭᖃᖅᑐᖅ ᖃᔅᓯᑦ ᐱᙳᐊᕈᑎ ᐊᑐᓂ ᐱᙳᐊᖅᑕᐅᓂᖓᓂᓗ ᐊᑐᓂ ᐊᓯᐅᔪᓐᓇᖅᑕᐅᓂᐊᕐᒪᖔᑕ ᐃᓱᒪᓕᐅᕐᓂᕐᒥᒃ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᐃᓱᖃᙱᑉᐸᑕ, ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅ “ᐊᑐᐊᒐᓚᒃᑐᖅ”, ᓱᕋᒃᑐᓐᓇᙱᒻᒪᑦ, ᑭᓯᐊᓂ ᐊᒻᒪᓗᑦᑐᖅ ᑭᒡᓕᖃᙱᑦᑐᒥᒃ ᐱᙳᐊᕈᑎᓕᐅᕐᓂᐊᖅᑐᖅ, ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᖅ ᑕᒪᑐᒪ ᑭᒡᓕᖃᙱᑦᑐᐊᑉ 2.70%-ᖓ. ᐱᙳᐊᕐᕕᐅᑉ ᑭᒡᓕᖓᐱᙳᐊᕐᕕᒃᐱᙳᐊᕐᕕᐅᑉ ᑭᒡᓕᖓᐱᙳᐊᕈᑎ ᒥᑭᓛᖅ ᐊᖏᓛᕐᓗ ᐊᖏᖅᑕᐅᔪᑦ. ᐊᖏᓛᖅ ᒪᕐᕉᕆᐊᖅᑎᑦᓯᔪᑦ ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᐅᔪᑦ ᓄᖅᑲᖅᑎᑦᑕᖏᑦ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᒫᑎᓐᒐᐃᓪᐸᕐᓇᐅᑎᒫᑎᓐᒐᐃᓪᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅ ᐊᓯᐅᔾᔨᓂᖅ ᑕᒪᐃᓐᓂ ᐱᙳᐊᕈᑎ ᒪᕐᕉᕆᐊᖅᑐᖅ. ᐸᕐᓇᐅᑏᑦ ᐃᓚᖓᓂ ᑕᑯᓗᒍ ᖃᓄᐃᓕᐅᕐᓂᖓ ᖃᓄᐃᓕᐅᙱᓐᓂᖓᓗ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᐊᑐᐊᒐᓚᙱᓐᓂᖓᓄᑦ ᐱᔾᔪᑎᐅᙱᑦᑐᖅ. ᓈᓴᐃᓂᖅ ᐱᔾᔪᑎᐅᔪᖅ.

ᒫᑎᓐᒐᐃᓪ $10-ᒥᑦ ᐱᒋᐊᕐᓗᓂ, 100 ᑲᐃᕙᕐᓃᑦ: ᑭᒡᓕᖓᑎᒍᑦ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᓴᖅᑭᑦᑐᖅ (ᐊᔾᔨᐅᖅᑕᐅᓂᖅ)$

$500 ᑭᒡᓕᖓ

≈ −$75

$1,000 ᑭᒡᓕᖓ

≈ −$75

$2,000 ᑭᒡᓕᖓ

≈ −$72

$5,000 ᑭᒡᓕᖓ

≈ −$70

ᐊᔾᔨᒌᒃᑐᖅ $10, ᖃᓄᐃᑦᑐᖅ ᑭᒡᓕᖓ

$27

04080

ᐊᑦᔨᑦ ᐊᓯᐅᔾᔨᓂᐅᑉ ᐊᖏᓂᖓᓂᒃ ᑕᑯᑎᑦᓯᔪᑦ. ᑭᒡᓕᖓ ᖁᑦᑎᓕᖅᑎᑕᐅᒐᓗᐊᖅᐸᑦ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᐊᓯᑦᑐᐊᓗᐊᙱᑦᑐᖅ; ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅ ᖃᓂᒋᔭᖓ $2,800-ᒥᒃ ᐱᙳᐊᕈᑎᓕᐅᖅᑐᖅ 100 ᑲᐃᕙᕐᓂᓄᑦ ᐊᑐᓂ, ᖁᓛ ᖃᓄᐃᑦᑐᐃᒃᑲᓗᐊᖅᐸᑦ.

ᐱᙳᐊᕐᕖᒃ ᒪᕐᕉᒃ ᓱᓕᔪᒃ

ᑑᕐᐲ (Thorp) ᐆᑦᑐᕋᐅᑎᖓ $1 ᐱᙳᐊᕐᕕᒃ, ᐊᖏᓛᖅ $1,000, ᐱᙳᐊᖅᑎ $30-ᒥᒃ ᓄᓇᖃᕐᓂᕐᒧᑦ ᐊᔪᒐᕈᒪᔪᖅ ᑕᐃᓚ ᐊᑕᐅᓯᕐᒥᒃ ᐊᑐᓂ. ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑦ 1, 2, 4 … 512 ᖁᓕᓂᒃ ᐱᙳᐊᕈᑎᓂᒃ ᐱᔪᓐᓇᖅᑎᑦᓯᔪᑦ, ᑕᐃᒪ ᖁᓕᑦ ᐊᓯᐅᔾᔨᓃᑦ ᑐᓪᓕᖅᑐᑦ ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅ ᓱᕋᒃᑎᑦᓯᔪᑦ: 785 ᐱᙳᐊᕈᓯᕐᓂ ᐊᑕᐅᓯᐊᖅ ᐃᐅᕌᐸ ᐊᒃᓴᓗᐊᖓᓂ (613-ᓂ ᐊᑕᐅᓯᐊᖅ ᐊᒥᐊᓕᑲ ᐊᒃᓴᓗᐊᖓᓂ), ᐊᑭᖓ $1,023. ᐱᙳᐊᕈᓰᑦ ᐱᖓᓱᑦ ᖁᓕᑦ ᐃᓘᓐᓇᑎᒃ ᐱᔭᕐᓂᖅᑐᑦ 96.2%-ᖓᓂ. ᐊᓯᖏᑦ 3.8% ᖃᓂᒋᔭᖓ $1,000-ᒥᒃ ᐊᓯᐅᔾᔨᔪᑦ, ᐅᓐᓄᐊᐅᑉ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᓴᖅᑭᑦᑐᖓ −$9.17, ᑕᒪᓐᓇ ᓱᓕᓪᓚᕆᒃᑐᒥᒃ $339-ᐅᑉ 1/37-ᖓ, ᐱᙳᐊᕈᓰᑦ ᐱᖓᓱᑦ ᖁᓕᑦ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᐱᙳᐊᕈᑎᒋᔭᖏᑦ. $5 ᐱᙳᐊᕐᕕᒃ ᐊᖏᓛᖅ $2,000 ᐊᔾᔨᒋᙱᑦᑐᖅ ᐃᒪᐃᙱᑦᑐᖅ: ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑦ $1,280-ᒧᑦ ᐊᕐᕕᓂᓖᑦ ᐱᖓᓱᓂᒃ ᐱᙳᐊᕈᑎᓂᒃ ᐱᔪᓐᓇᖅᑎᑦᓯᔪᑦ, ᓱᕋᒃᑐᓗ 403 ᐱᙳᐊᕈᓯᕐᓂ ᐊᑕᐅᓯᐊᖅ ᐃᐅᕌᐸ ᐊᒃᓴᓗᐊᖓᓂ (323-ᓂ ᐊᑕᐅᓯᐊᖅ ᐊᒥᐊᓕᑲ ᐊᒃᓴᓗᐊᖓᓂ) ᐊᑭᖃᖅᑐᓗ $2,555.

ᒪᕐᕉᕆᐊᖅᑎᑕᐅᔪᑦ ᐱᙳᐊᕈᑏᑦ ᐱᙳᐊᕐᕕᐅᑉ ᐊᖏᓛᖅ ᑭᒡᓕᖓᓄᑦ ᖁᑦᑎᒋᐊᖅᑐᑦ, ᓂᕿᓗ ᐅᓐᓄᐊᖅᓯᐅᑎ ᓴᓂᐊᓂ ᐃᒡᓗᓯᐅᑎᒥ ᐅᑕᖅᑭᔪᖅ.

ᐱᙳᐊᖅᑎ ᑮᓇᐅᔭᖃᖅᑐᐊᓗᒃ

ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᕐᒧᑦ ᐱᙳᐊᕈᑎᒃᓴᓂᒃ ᐊᒥᓱᐊᓗᓐᓂᒃ ᑐᓂᓯᓗᑎᑦ ᒥᑭᔪᒥᒃ ᑭᒡᓕᖃᖅᑎᑦᑎᓇᑎᑦ: 1,000 ᐊᑦᔨᐅᑏᑦ, ᐱᒋᐊᕐᕕᒃ ᐊᑦᔨᐅᑎ ᐊᑕᐅᓯᖅ, 1,000 ᑲᐃᕙᕐᓃᓪᓗ ᐊᐅᐸᓗᒃᑐᒥ, ᐊᑦᔨᐅᑎᒥᑦ ᐊᑕᐅᓯᕐᒥᑦ ᐱᒋᐊᖅᑭᓪᓗᓂ ᑐᓪᓕᐊ ᒪᕐᕉᕆᐊᖅᑎᑦᓯᓂᖅ ᐱᑕᖃᖅᑐᓂᑦ ᐊᖏᓂᖅᓴᐅᒃᐸᑦ. ᒥᓕᐊᓐ ᐊᑕᐅᓯᖅ ᐅᓐᓄᐊᑦ ᐊᔾᔨᐅᖅᑕᐅᓂᖏᓐᓂᒃ ᐊᐅᓚᑦᑎᓯᒪᔪᒍᑦ. ᐃᐅᕌᐸ ᐊᒃᓴᓗᐊᖓᓂ 52% ᑮᓇᐅᔭᒃᓴᖅᑐᑦ ᐃᓱᓕᑦᑐᑦ, 21% ᐃᓘᓐᓇᑎᒃ ᐊᓯᐅᔾᔨᔪᑦ, ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᓴᖅᑭᑦᑐᖅ −124 ᐊᑦᔨᐅᑏᑦ. ᐱᙳᐊᕐᓂᕐᒥ ᓯᕉᖃᙱᑦᑐᒥᐊᒃᓴᓗᐊᖅᓯᕉᐊᑕᐅᓯᖅ ᐅᔾᔪᐊᕌᔭᒃ ᐃᓂ ᐊᔾᔨᒌᒃᑐᒥᒃ ᐊᑭᓕᐅᑎᓕᖕᓄᑦ, ᑕᓴᓐᓄᑦ ᓇᑉᐸᖅᑐᓄᓪᓘᓐᓃᑦ ᐱᙳᐊᕈᑏᑦ ᐃᓚᐅᑎᙱᑕᖓ; ᑳᓯᓅᑉ ᐃᓚᖓᑕ ᐱᓇᓪᓗᑎᖓ. ᐊᓯᐊᓂ ᐊᑎᖏᑦ: ᐊᑕᐅᓯᖅ ᓯᕉ, 0.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᐊᓪᓛᑦ, ᐱᙳᐊᕈᑏᑦ ᐃᓘᓐᓇᑎᒃ ᓈᒻᒪᒃᑎᓪᓗᒋᑦ, ᖃᓂᒋᔭᖓ 60% ᑐᐊᖅ ᑮᓇᐅᔭᒃᓴᖅᑐᑦ ᐃᓱᓕᓐᓂᐊᖅᑐᑦ. ᐅᓐᓄᐊᖅ ᐊᑐᖅᑕᐅᒐᔪᒃᑐᖅ ᒥᑭᔪᒥᒃ ᐊᔪᒐᕐᓂᖃᖅᑐᖅ; ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᐊᑭᓕᖅᑕᐅᔪᖅ ᐅᓐᓄᐊᓄᑦ ᓯᕉᒧᑦ ᐃᓱᓕᑦᑐᓄᑦ.

ᐊᒃᓴᓗᐊᖅᑮᓇᐅᔭᒃᓴᖅᑐᑦ ᐃᓱᓕᑦᑐᑦ1,000 ᐊᑦᔨᐅᑏᑦ ᐊᓯᐅᔾᔨᔪᑦᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᓴᖅᑭᑦᑐᖅ

ᓯᕉᖃᙱᑦᑐᖅ (ᓈᒻᒪᒃᑐᖅ ᐱᙳᐊᕐᓂᖅ)60%15%0

ᐃᐅᕌᐸ52%21%−124 ᐊᑦᔨᐅᑏᑦ

ᐊᒥᐊᓕᑲ45%30%−250 ᐊᑦᔨᐅᑏᑦ

ᐊᔾᔨᐅᖅᑕᐅᓂᕗᑦ, ᒥᓕᐊᓐ ᐊᑕᐅᓯᖅ ᐅᓐᓄᐊᑦ ᐊᒃᓴᓗᐊᕐᒧᑦ ᐊᑐᓂ. ᐊᔾᔨᒌᒃᑎᑕᐅᔪᑦ 1/37 1/19-ᓗ ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᐅᑉ 1,000 ᑲᐃᕙᕐᓂᓂ ᐱᙳᐊᕈᑎᒋᔭᖏᓐᓂᑦ: ᖃᓂᒋᔭᖓ 4,600 ᐊᑦᔨᐅᑏᑦ ᐃᐅᕌᐸ ᐊᒃᓴᓗᐊᖓᓂ, 4,750-ᓗ ᐊᒥᐊᓕᑲ ᐊᒃᓴᓗᐊᖓᓂ.

ᐃᓕᓐᓂᐊᒃᓴᖅ

ᑳᓯᓄᐃᑦ ᑭᒡᓕᓕᐅᖅᐸᒃᑐᑦ ᓇᒻᒥᓂᖅ ᐊᔾᔨᒌᙱᓐᓂᖓᑦᓈᓴᐃᓂᖅᐊᔾᔨᒌᙱᓐᓂᖅᐱᓐᓂᐅᔪᑦ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᑭᖓᑕ ᐊᕙᓗᐊᓂ ᖃᓄᖅ ᓯᐊᒻᒪᒃᓯᒪᓂᖏᓐᓂᒃ ᓈᓴᐅᑎ. ᐊᑕᐅᓯᕐᒧᑦ ᐱᙳᐊᕈᑎᒥ ᐊᖏᔪᖅ, ᐊᔾᔨᒌᒃᑐᒥᒃ ᐊᑭᓕᐅᑎᓕᖕᓂ ᐱᙳᐊᕈᑏᓐᓂ ᒥᑭᔪᖅ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᑭᒡᓕᖃᖅᑎᓐᓂᐊᕐᓗᒋᑦ, ᐸᕐᓇᐅᑏᑦ ᐊᔪᐃᓐᓇᖅᑎᓐᓂᐊᕐᓗᒋᑦ ᐃᒪᐃᙱᑦᑐᖅ. ᒫᑎᓐᒐᐃᓪ ᐊᔪᐃᓐᓇᖅᑎᓐᓂᖓ ᖁᕕᐊᒋᔭᖓ ᐃᒪᐃᓐᓂᐊᙱᑦᑐᖅ, ᓈᓴᐃᓂᐅᓪᓗ ᑕᐃᒪᐃᑦᑐᖅ ᐊᔪᐃᓐᓇᖅᑎᑦᓯᓂᐊᕐᒪᑦ. ᐱᙳᐊᖅᑎ ᑭᒡᓕᖓ ᐊᐱᖅᑯᑎᐅᓂᖓᓂᒃ ᐃᓱᒪᔪᖅ ᐃᖏᕐᕋᓂᖅᓴᒥᒃ ᓇᓂᓯᓯᒪᔪᖅ, ᑭᓯᐊᓂ ᖃᖅᑲᖅ ᐃᖏᕐᕋᓐᓂᐊᕐᓂᖓᓐᓂ ᑕᒻᒪᖅᑐᖅ.

ᓈᓴᐅᑎᓕᐅᕈᑎ

p(ᓱᕋᒃᑐᖅ) = (19/37)ᵏ, k = ᑭᒡᓕᐅᑉ ᐱᔪᓐᓇᖅᑎᑦᓯᔭᖏᑦ ᐊᓯᐅᔾᔨᓃᑦ

ᓱᕋᒃᑎᓪᓗᒍ ᐊᓯᐅᔾᔨᓂᖅ = ᐱᒋᐊᕐᕕᒃ · (2ᵏ − 1)

E[ᐱᙳᐊᕈᓯᕐᒧᑦ ᐊᓯᐅᔾᔨᓂᖅ] = E[ᐱᙳᐊᕈᓯᕐᒥ ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑦ] / 37

ᐊᐱᖅᑯᑏᑦ

ᐱᙳᐊᕈᑎᒃᓴᐃᑦ ᐊᖏᓂᖅᓴᐃᑦ ᓈᒻᒪᒃᓴᖅᑐᓐᓇᖅᐸᑦ?

ᖁᑦᑎᓂᖅᓴᒧᑦ ᑭᒡᓕᖓᓄᑦ ᑎᑭᑦᑎᔪᓐᓇᖅᑎᑦᓯᔪᑦ, ᓱᕋᒃᑐᖅ ᐃᒪᐃᓐᓂᖅᓴᐅᓪᓗᓂ ᐊᖏᓂᖅᓴᐅᓪᓗᓂᓗ. ᑕᐃᓚᒧᑦ ᓂᕆᐅᒋᔭᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᖅ ᐊᓯᑦᑐᙱᑦᑐᖅ.

“ᐊᖏᔪᖅ ᒫᑎᓐᒐᐃᓪ” ᐊᑦᔨᐅᑎᒥᒃ ᐃᓚᒋᐊᖅᓯᔪᖅ ᒪᕐᕉᕆᐊᖅᓯᓂᕐᒥ ᐊᑐᓂ ᖃᓄᐃᑦᑐᐊ?

ᐱᙳᐊᕈᑏᑦ ᑮᓇᐅᔭᖏᑦ 10, 30, 70, 150, 310: $500 ᑭᒡᓕᖓ ᑕᓪᓕᒪᑦ ᐊᓯᐅᔾᔨᓃᑦ ᑭᖑᓪᓗᒋᑦ ᑎᑭᑕᐅᔪᖅ, ᓱᕋᒃᑐᓗ $570-ᒥᒃ ᐊᑭᖃᖅᑐᖅ. ᐊᔾᔨᒋᔭᖓ 2.70%.

ᓱᒻᒪᑦ ᑳᓯᓄᐃᑦ ᐊᖏᓛᒥᒃ ᑭᒡᓕᖃᖅᐸᑦ?

ᐱᙳᐊᖅᑎ ᐊᑕᐅᓯᖅ ᑳᓯᓅᑉ ᑐᖁᔭᖓᑕ ᐊᑭᓕᑦᑎᐊᕈᓐᓇᖅᑕᖓᓂᑦ ᐊᒥᓲᓂᖅᓴᓂᒃ ᐱᙳᐊᕈᑎᓕᐅᕈᓐᓇᙱᑦᑐᖅ. ᐊᔾᔨᒌᙱᓐᓂᕐᒥᒃ ᐊᐅᓚᑦᓯᓂᖅ.

ᓈᓴᐃᓂᕐᒥᒃ ᐱᙳᐊᕐᓂᖅ

ᓯᕗᓪᓕᐅᓪᓗᓂ ᐃᓕᖕᓂᒃ ᐆᒃᑐᕐᓗᑎᑦ. ᑭᐅᔾᔪᑏᑦ ᐃᓗᓕᖏᑦ ᒪᑐᔭᐅᓯᒪᔪᑦ; ᐅᐸᒍᓐᓇᕈᕕᑦ ᒪᑐᐃᖅᓯᓗᒍ.

ᒫᑎᓐᒐᐃᓪᐸᕐᓇᐅᑎᒫᑎᓐᒐᐃᓪᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᖅ ᐊᓯᐅᔾᔨᓂᖅ ᑕᒪᐃᓐᓂ ᐱᙳᐊᕈᑎ ᒪᕐᕉᕆᐊᖅᑐᖅ. ᐸᕐᓇᐅᑏᑦ ᐃᓚᖓᓂ ᑕᑯᓗᒍ ᖃᓄᐃᓕᐅᕐᓂᖓ ᖃᓄᐃᓕᐅᙱᓐᓂᖓᓗ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → $100-ᒥᑦ ᐊᐅᐸᓗᒃᑐᒧᑦᐱᙳᐊᕈᑎᐊᐅᐸᓗᒃᑐᖅᐊᔾᔨᒌᒃᑐᒥᒃ ᐊᑭᓕᐅᑎᓕᒃ ᐱᙳᐊᕈᑎ 18 ᐊᐅᐸᓗᒃᑐᓄᑦ ᓈᓴᐅᑎᓄᑦ. ᕖᓇᓯᑐᑦ: ᕈᔾ (rouge).ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᐱᒋᐊᖅᑐᖅ, ᐃᒻᒥᓂᓗ n ᐱᙳᐊᕈᑏᓐᓂᒃ ᐅᖓᑖᓄᑦ ᐱᙳᐊᕈᓯᕐᒥ ᐱᔪᓐᓇᖅᑎᑦᓯᙱᑦᑐᖅ: n ᐊᓯᐅᔾᔨᓃᑦ ᑐᓪᓕᖅᑐᑦ ᐊᓂᒍᖅᑎᓪᓗᒋᑦ ᓄᖅᑲᖅᓗᓂ ᐱᒋᐊᖅᑲᐃᔪᖅ. ᐱᙳᐊᕈᓯᐅᑉ ᐊᑕᐅᓯᐅᑉ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᓴᖅᑭᑦᑕᖓ n-ᒥ ᓈᓴᐅᑎᓕᐅᕈᑎᑐᑦ ᑎᑎᕋᕐᓗᒍ, ᓈᓴᐅᑎᓗᒍᓗ 5, 6 7-ᓗ ᐱᙳᐊᕈᑏᓐᓄᑦ ᐃᐅᕌᐸ ᐊᒃᓴᓗᐊᖓᓂ.
ᑭᐅᔾᔪᑎ ᑕᑯᓗᒍ
ᑭᐅᔾᔪᑎ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ 100 × (1 − (38/37)ⁿ): −$14.26 5 ᐱᙳᐊᕈᑏᓐᓄᑦ, −$17.35 6-ᓄᑦ −$20.52-ᓗ 7-ᓄᑦ. ᐱᙳᐊᕈᑏᑦ ᐊᒥᓲᓂᖅᓴᐅᑎᓪᓗᒋᑦ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ ᐱᙳᐊᕈᓯᖅ ᐱᐅᓂᖅᓴᐅᙱᑦᑐᖅ, ᐱᐅᓐᖏᓂᖅᓴᐅᓕᖅᑐᖅ.
ᐱᙳᐊᕈᓯᖅ $100-ᒥᒃ ᐊᔪᒐᖅᑐᖅ n ᐱᙳᐊᕈᑏᑦ ᑕᒪᕐᒥᒃ ᐊᓯᐅᔾᔨᙱᑉᐸᑕ, ᑕᒪᓐᓇ ᐱᒋᐊᖅᑐᖅ ᐱᔪᓐᓇᕐᓂᓕᒃ qⁿ, q = 19/37. ᑕᒪᕐᒥᒃ ᐊᓯᐅᔾᔨᑉᐸᑕ, ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖏᑦ 100, 200 … 100 × 2ⁿ⁻¹ ᑲᑎᑕᐅᓪᓗᑎᒃ 100 × (2ⁿ − 1). ᑕᐃᒪᐃᒻᒪᑦ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᖅ 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
ᑕᒫᓂ 2q = 38/37, 1-ᒥᑦ ᐊᖏᓂᖅᓴᐅᑐᐊᖅ, ᑕᐃᒪᐃᒻᒪᑦ (2q)ⁿ n-ᒥᒃ ᐊᖏᒡᓕᕙᓪᓕᐊᔪᖅ ᐊᔾᔨᒌᒃᑎᑕᐅᔪᓗ ᒥᑭᓕᕐᐳᖅ: n = 5 1.1426-ᒥᒃ −$14.26-ᒥᒡᓗ ᓴᖅᑭᑦᑎᔪᖅ; n = 6 1.1735-ᒥᒃ −$17.35-ᒥᒡᓗ; n = 7 1.2052-ᒥᒃ −$20.52-ᒥᒡᓗ. ᐊᑦᔨᐅᑏᑦ ᑕᑭᓂᖅᓴᐃᑦ ᐊᓯᐅᔾᔨᕙᓐᖏᓂᖅᓴᐅᔪᑦ, ᐱᙳᐊᕈᓯᐅᔪᑦ 3.6%, 1.8% 0.9%-ᖏᓐᓂᓗ, ᑭᓯᐊᓂ ᐊᓯᐅᔾᔨᓂᖅ ᑕᒪᐃᓐᓇ ᒪᕐᕉᕆᐊᖅᓗᒍ ᐊᖏᓂᖅᓴᐅᔪᖅ, ᑮᓇᐅᔭᓪᓗ ᐊᒥᓲᓂᖅᓴᐃᑦ ᐱᙳᐊᕐᕕᒃ ᖄᖏᖅᐸᒃᑕᖏᑦ ᐱᙳᐊᕈᓯᕐᒥ ᑕᒪᐃᓐᓂ, ᐱᙳᐊᕈᑎᐅᑉ ᑕᒪᐃᓐᓇᑕ 2.70%-ᖓᓂᒃ ᐊᑭᓕᓪᓗᑎᒃ.
ᐊᒥᐊᓕᑲ ᐊᒃᓴᓗᐊᖓᓂ 2q = 40/38 ᓈᓴᐅᑏᓪᓗ −$29.24, −$36.04 −$43.20-ᓗ (ᑎᐃᓗ, Taylor, 2021). ᓯᕉᐊᒃᓴᓗᐊᖅᓯᕉᐊᑕᐅᓯᖅ ᐅᔾᔪᐊᕌᔭᒃ ᐃᓂ ᐊᔾᔨᒌᒃᑐᒥᒃ ᐊᑭᓕᐅᑎᓕᖕᓄᑦ, ᑕᓴᓐᓄᑦ ᓇᑉᐸᖅᑐᓄᓪᓘᓐᓃᑦ ᐱᙳᐊᕈᑏᑦ ᐃᓚᐅᑎᙱᑕᖓ; ᑳᓯᓅᑉ ᐃᓚᖓᑕ ᐱᓇᓪᓗᑎᖓ. ᐊᓯᐊᓂ ᐊᑎᖏᑦ: ᐊᑕᐅᓯᖅ ᓯᕉ, 0.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ →ᖃᙱᑦᑐᒥ ᐱᙳᐊᕐᓂᕐᒥ 2q = 1, ᐱᙳᐊᕈᓯᕐᓗ ᑕᒪᐃᓐᓇ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᓯᕉ ᓱᓕᓪᓚᕆᒃ, ᐊᑦᔨᐅᑏᑦ ᖃᓄᑐᐃᓐᓇᖅ ᑕᑭᑎᒋᒃᑲᓗᐊᖅᐸᑕ.
“ᐊᖏᔪᖅ ᒫᑎᓐᒐᐃᓪ” ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖓᓂᒃ ᐊᓯᐅᔾᔨᓂᖅ ᐊᓂᒍᖅᑎᓪᓗᒍ ᒪᕐᕉᕆᐊᖅᓯᔪᖅ ᐊᑦᔨᐅᑎᒥᒡᓗ ᐊᑕᐅᓯᕐᒥᒃ ᐊᑦᔨᐅᑎᒥᒃᑮᓇᐅᔭᖅᐊᑦᔨᐅᑎᐸᕐᓇᐅᑎᐅᑉ ᐅᕝᕙᓘᓐᓃᑦ ᐱᙳᐊᖅᑎᐅᑉ ᑐᙵᕕᒋᔭᖓ ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖓ. ᐊᓯᑦᑐᖅᐸᓪᓕᐊᓂᐅᔪᑦ ᐊᑦᔨᐅᑎᓄᑦ ᓇᓗᓇᐃᖅᑕᐅᕙᒃᑐᑦ, ᑕᐃᒪ ᓇᓕᐊᓂᑐᐃᓐᓇᖅ ᐱᙳᐊᕐᕕᒻᒥ ᐊᑐᕈᓐᓇᖁᓪᓗᒋᑦ.ᐃᓘᓐᓇᖓ ᐅᖃᓕᒫᕐᓗᒍ → ᐃᓚᒋᐊᖅᓯᓪᓗᓂ, ᑕᐃᒪᐃᒻᒪᑦ ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖏᑦ 1, 3, 7, 15, 31 … ᓯᓚᑦᑐᐃᔪᑦ ᐊᔪᒐᕐᓂᖅ ᓯᕗᓪᓕᖅᐹᖅ 1-ᒥ, 2-ᒥ, 3-ᒥ ᐅᕝᕙᓘᓐᓃᑦ k-ᒥ ᐱᙳᐊᕈᑎᒥ ᓴᖅᑭᑉᐸᑦ ᖃᔅᓯᓂᒃ ᐊᔪᒐᖅᓯᒪᕖᑦ? $10 ᐊᑦᔨᐅᑎᒥ $500-ᓗ ᐊᖏᓛᖑᔪᒥ, ᐱᙳᐊᕈᓯᖅ ᖃᓄᖅ ᐊᒥᓱᓪᓚᕆᒃᑐᓂ ᓱᕋᒃᐸᑦ, ᐱᙳᐊᕈᓯᕐᓗ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᖃᔅᓯᓂᒃ ᐊᑭᖃᖅᐸ?
ᑭᐅᔾᔪᑎ ᑕᑯᓗᒍ
ᑭᐅᔾᔪᑎ ᐊᔪᒐᕐᓂᖅ k-ᒥ ᐱᙳᐊᕈᑎᒥ k ᐊᑦᔨᐅᑏᓐᓂᒃ ᐊᔪᒐᖅᓯᒪᑎᑦᓯᔪᖅ: +1, +2, +3 ᐊᓯᖏᓪᓗ. $10-ᒥ $500-ᒥᓗ ᐱᙳᐊᕈᓯᖅ ᓱᕋᒃᑐᖅ ᑕᓪᓕᒪᑦ ᐊᓯᐅᔾᔨᓃᑦ ᐊᓂᒍᖅᑎᓪᓗᒋᑦ, ᐱᙳᐊᕈᓯᐅᔪᑦ 3.57%-ᖏᓐᓂ, −$570-ᒧᑦ. ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᐱᙳᐊᕈᓯᖅ $2.32-ᒥᒃ ᐊᓯᐅᔾᔨᔪᖅ $85.73 ᐱᙳᐊᕈᑎᒋᔭᐅᔪᓂᑦ: 2.70%.
k-ᖓ ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖓ 2ᵏ − 1 ᐊᑦᔨᐅᑏᑦ: 1, 3, 7, 15, 31. ᓯᕗᓪᓕᐅᔪᑦ k − 1 ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖏᑦ ᑲᑎᑕᐅᓪᓗᑎᒃ 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. k-ᖓᓂ ᐱᙳᐊᕈᑎᒥ ᐊᔪᒐᕐᓂᖅ 2ᵏ − 1-ᒥᒃ ᐊᑭᓕᐅᑎᖃᖅᑐᖅ, ᑕᐃᒪᐃᒻᒪᑦ ᐱᙳᐊᕈᓯᖅ ᐃᓱᓕᑦᑐᖅ (2ᵏ − 1) − (2ᵏ − k − 1) = k ᐊᑦᔨᐅᑏᑦ ᐊᔪᒐᖅᓯᒪᓪᓗᑎᑦ: ᐊᑦᔨᐅᑎ ᐊᑕᐅᓯᖅ ᐃᓚᒋᐊᖅᑕᐅᔪᖅ ᐊᓯᐅᔾᔨᓂᕐᒧᑦ ᑕᒪᐃᓐᓄᑦ ᐊᔪᒐᕐᓂᖅ ᓯᕗᓂᐊᒍᑦ.
$10 ᐊᑦᔨᐅᑎᒥ ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖏᑦ $10, $30, $70, $150 $310-ᓗ; ᑐᓪᓕᐊ, $630, $500 ᐊᖏᓛᖑᔪᒥᑦ ᐊᖏᓂᖅᓴᐅᔪᖅ. ᑕᓪᓕᒪᑦ ᐊᓯᐅᔾᔨᓃᑦ ᑐᓪᓕᖅᑐᑦ, (19/37)⁵ = 3.57%, ᐊᑭᖃᖅᑐᑦ 10 + 30 + 70 + 150 + 310 = $570.
ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ: ᐊᔪᒐᕐᓃᑦ $10 × k-ᒥᒃ ᐱᔪᖃᖅᑎᑦᓯᔪᑦ ᐱᔪᓐᓇᕐᓂᓕᒃ (19/37)ᵏ⁻¹ × 18/37 k = 1-ᒥᑦ 5-ᒧᑦ, ᓱᕋᒃᑐᓗ $570-ᒥᒃ ᐱᔪᖅ ᐱᔪᓐᓇᕐᓂᓕᒃ 3.57%; ᐊᑕᐅᑦᑎᒃᑯᑦ −$2.32. ᑮᓇᐅᔭᑦ ᐱᙳᐊᕈᑎᒋᔭᐅᔪᑦ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ $85.73 ᐱᙳᐊᕈᓯᕐᒥ, 2.32/85.73 = 2.70%-ᓗ. ᐊᑐᖅᑕᐅᓂᖅᐹᖅ ᒫᑎᓐᒐᐃᓪ ᐊᔾᔨᒋᔭᐅᔪᒥ ᐱᙳᐊᕐᕕᖕᒥ ᓱᕋᒃᑐᖅ ᐱᖓᓲᔪᖅᑐᑦ ᐊᓯᐅᔾᔨᓃᑦ ᐊᓂᒍᖅᑎᓪᓗᒋᑦ, ᐱᙳᐊᕈᓯᐅᔪᑦ 1.83%-ᖏᓐᓂ, −$630-ᒧᑦ, $1.74-ᒥᒡᓗ ᐊᓯᐅᔾᔨᔪᖅ $64.20 ᐱᙳᐊᕈᑎᒋᔭᐅᔪᓂᑦ: 2.70% ᐊᒃᑲᓐᓂ.
ᐱᙳᐊᖅᑎ ᒫᑎᓐᒐᐃᓪᒥᒃ ᐊᐅᐸᓗᒃᑐᒧᑦ ᐸᕐᓇᐃᔪᖅ $1 ᓯᕗᓪᓕᖅᐹᒥ ᐱᙳᐊᕈᑎᒋᓪᓗᒍ ᐱᙳᐊᕐᕕᖕᒥ $1,000 ᑭᒡᓕᓕᖕᒥ, ᑕᐃᒪᐃᒻᒪᑦ ᖁᓕᑦ ᐱᙳᐊᕈᑏᑦ ᐱᔪᓐᓇᖅᑐᑦ: $1, $2, $4 ᐊᓯᖏᓪᓗ $512-ᒧᑦ. ᐱᙳᐊᕈᓯᖅ ᖃᓄᖅ ᐊᒥᓱᓪᓚᕆᒃᑐᓂ ᖁᓕᑦ ᑕᒪᕐᒥᒃ ᐊᓯᐅᔾᔨᕙᑦ, ᑕᒪᓐᓇ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᖃᓄᖅ ᐊᑯᓂᐅᑎᒋᕙ, ᐅᓐᓄᒃᑯᑦ 200 ᐱᙳᐊᕈᑏᑦ ᖃᓄᖅ ᐊᒥᓱᓪᓚᕆᒃᑐᓂ ᑕᐃᒪᐃᓕᖓᓗᑎᒃ ᐃᓱᓕᑦᑕᕐᐸᑦ?
ᑭᐅᔾᔪᑎ ᑕᑯᓗᒍ
ᑭᐅᔾᔪᑎ ᐱᙳᐊᕈᓯᖅ ᖁᓕᑦ ᑕᒪᕐᒥᒃ ᐊᓯᐅᔾᔨᔪᖅ ᖃᓂᒋᔭᖓ 784 ᐃᓚᖏᓐᓂ ᐊᑕᐅᓯᕐᒥ, $1,023-ᒥᒃ ᐊᑭᖃᖅᑐᖅ. ᑕᒪᓐᓇ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ ᖃᓂᒋᔭᖓ 1,600 ᐱᙳᐊᕈᑏᑦ, ᐅᓐᓄᒃᑯᓪᓗ 200 ᐱᙳᐊᕈᑏᑦ ᑭᒡᓕᖓᓄᑦ ᑎᑭᑦᑐᑦ ᖃᓂᒋᔭᖓ 11% ᐱᕕᖏᓐᓂ. ᐱᙳᐊᕈᓯᖅ ᓱᓕ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ $0.31-ᒥᒃ ᐊᓯᐅᔾᔨᔪᖅ $11.31 ᐱᙳᐊᕈᑎᒋᔭᐅᔪᓂᑦ: 2.70%.
ᐊᐅᐸᓗᒃᑐᒧᑦ ᐱᙳᐊᕈᑎ ᐊᓯᐅᔾᔨᔪᖅ ᐱᔪᓐᓇᕐᓂᓕᒃ 19/37, ᓯᕉ ᐃᓚᐅᓪᓗᓂ. ᖁᓕᑦ ᑐᓪᓕᖅᑐᑦ: (19/37)¹⁰ ≈ 0.00128, ᖃᓂᒋᔭᖓ 784 ᐃᓚᖏᓐᓂ 1. ᐱᙳᐊᕈᑎ ᑮᓇᐅᔭᖏᑦ 1 + 2 + 4 + … + 512 ᑲᑎᑕᐅᓪᓗᑎᒃ $1,023.
ᐱᙳᐊᕈᓯᖅ ᐊᔾᔨᒌᒃᑎᓪᓗᒋᑦ 2.05 ᐱᙳᐊᕈᑏᓐᓂᒃ ᐱᔭᕆᐊᖃᖅᑐᖅ, ᑕᐃᒪᐃᒻᒪᑦ 784 ᐱᙳᐊᕈᓯᐅᔪᑦ ᖃᓂᒋᔭᖓ 1,600 ᐱᙳᐊᕈᑏᓐᓂᒃ ᐱᔭᕆᐊᖃᖅᑐᑦ. ᑲᐃᕙᕐᓂᖅ ᑲᐃᕙᕐᓂᕐᒧᑦ ᓈᓴᖅᑕᐅᓪᓗᑎᒃ, ᖁᓕᑦ ᐊᓯᐅᔾᔨᓃᑦ ᑐᓪᓕᖅᑐᑦ 200 ᐱᙳᐊᕈᑏᓐᓂ ᓇᒧᑐᐃᓐᓇᖅ ᓴᖅᑭᑦᑐᑦ 11.3% ᐱᕕᖏᓐᓂ.
ᐊᑦᑕᓇᖅᑐᑦ ᐊᑯᕐᖏᓐᓂ ᐱᙳᐊᖅᑎ $1-ᒥᒃ ᐱᙳᐊᕈᓯᕐᒥ ᐊᔪᒐᖅᑐᖅ: ᖃᓂᒋᔭᖓ $783 ᐱᐅᙱᑦᑐᖅ ᓯᕗᓂᐊᒍᑦ, ᑕᐃᑦᓱᒪᓂᓗ −$1,023, ᖃᓂᒋᔭᖓ −$240 ᑕᒪᐃᓐᓂ. ᐱᙳᐊᕈᓯᕐᒥ: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. ᑮᓇᐅᔭᑦ ᐱᙳᐊᕈᑎᒋᔭᐅᔪᑦ ᐱᙳᐊᕈᓯᕐᒥ: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. −0.31/11.31 = −2.70%-ᓗ.
ᐊᔾᔨᒋᙱᑦᑐᖓ: $10 ᓯᕗᓪᓕᖅᐹᖅ ᐱᙳᐊᕈᑎ $500 ᑭᒡᓕᓕᖕᒥ, ᐱᖓᓲᔪᖅᑐᑦ ᐱᙳᐊᕈᑏᑦ: ᑭᒡᓕᖓ 55 ᐱᙳᐊᕈᓯᐅᔪᑦ ᐃᓚᖏᓐᓂ ᐊᑕᐅᓯᕐᒥ ᑎᑭᑕᐅᔪᖅ, ᐅᓐᓄᒃᑯᓪᓗ 200 ᐱᙳᐊᕈᑏᑦ 84% ᐱᕕᖏᓐᓂ ᑎᑭᑦᑐᑦ. ᑭᒡᓕᖓ ᐊᖏᓂᖅᓴᖅ ᓱᕋᒃᑐᒥᒃ ᐱᒋᐊᕐᓂᖏᓐᓂᖅᓴᐅᑎᑦᓯᔪᖅ ᐊᖏᓂᖅᓴᐅᑎᑦᓯᓪᓗᓂᓗ, ᐊᑭᑭᓐᓂᖅᓴᐅᑎᑦᓯᙱᑦᑐᖅ.

ᐃᓪᓗᕐᓇᖅᑐᖅ ᐊᔾᔨᓕᐊᖅ

ᐃᒪᐃᓐᓂᖅᓴᐅᓪᓗᓂ ᐊᖏᓂᖅᓴᐅᓪᓗᓂᓗ, ᑭᒡᓕᖏᓐᓂ ᑕᒪᐃᓐᓂ. ᐊᐱᕆᔭᐅᔪᖅ ᐊᓯᑦᑐᐃᓐᓇᙱᑦᑐᖅ.

ᐱᕝᕕᒋᔭᐅᔪᑦ

Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. ᐃᓚᖓ 13, ᕉᓚᑦ.
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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