Asɛmmisa: ɛpono ahyeɛ
$10 Martingale wɔ $500 ɛpono so. Mpɛn ahe na ɛbubu, sika ahe na ɛhwere bere a ɛbubu, na sɛ wode ahyeɛ no kɔ soro a, ɛboa?
Ahwereɛ nsia a ɛdi so bubu no: 1.8% wɔ ɔkwan no kyinhyia mua biara mu, bɛyɛ 60% wɔ agorɔ bere a kyinhyia 100 wom so. Mmubuiɛ no bɔ $630 ka. Sɛ wode ahyeɛ no kɔ $5,000 so a, mmubuiɛ no mmaa mma (0.25% wɔ kyinhyia mua biara mu) na ɛyɛ kɛseɛ ($5,110). Ahwereɛ a wɔhwɛ kwan wɔ dɔla biara a wode to so yɛ 2.70% wɔ ahyeɛ biara so.
Asɛmmisa no
Agorɔfoɔ bi de $10 to kɔkɔɔbɛtKɔkɔɔSika pɛ bɛt wɔ nɔma kɔkɔɔ 18 no so. Franse kasa mu: rouge.Kenkan ne nyinaa → so na ɔyɛ no mmɔho mmienu wɔ ahwereɛ biara akyi, na ɔsan kɔ ahyɛaseɛ wɔ nkonim akyi. Ɛpono ahyeɛ soro yɛ $500. (a) Ahwereɛ ahe a ɛdi so na ɛbubu bɛt nsakraeɛ ntoatoasoɔɔkwanBɛt nsakraeɛ ntoatoasoɔMmara biara a ɛsesa bɛt sika no gyina deɛ atwam so. Ntoatoasoɔ a ɛkɔ soro ahwereɛ akyi ma no kɔ soro ahwereɛ akyi, na deɛ ɛkɔ soro nkonim akyi ma no kɔ soro nkonim akyi.Kenkan ne nyinaa → no? (b) HokwanakontaabuHokwanHokwan a biribi bɛba, firi 0 kɔsi 1 anaa 0% kɔsi 100%. Nɔma baako: 1/37 = 2.70%.Kenkan ne nyinaa → no yɛ ahe wɔ kyinhyia mua biara mu? (c) Ɛdeɛn na ɛhwere wɔ mmubuiɛ no mu? (d) Ɛkwan bɛn so na $5,000 ahyeɛ sesa mmuaeɛ no? (e) Ahyeɛ bi wɔ hɔ a ɛma nhyehyɛe no nya mfasoɔ?
Mmuaeɛ (a)–(c)
Bɛt sika: 10, 20, 40, 80, 160, 320. Deɛ ɛtɔ so nson bɛyɛ 640 > 500, ɛno nti ahwereɛ nsia bubu no. (b) p(ahwereɛ) = 19/37 wɔ kɔkɔɔ so; (19/37)⁶ = 0.0183, bɛyɛ 1.8% wɔ kyinhyia mua biara mu. Kyinhyia mua baako kɔ bɛyɛ kyinhyia mmienu wɔ mfinimfini mu, ɛno nti agorɔ bere a kyinhyia 100 wom no wɔ bɛyɛ kyinhyia mua 50 na p(mmubuiɛ baako anaa deɛ ɛboro saa) ≈ 1 − 0.9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, a ɛyɛ ahyɛaseɛ kyɛfasikaKyɛfaBɛt sika a ɛwɔ ase a nhyehyɛe anaa agorɔfoɔ de yɛ adwuma. Wɔka bɛt nsakraeɛ ntoatoasoɔ ho asɛm wɔ kyɛfa mu sɛdeɛ ɛbɛyɛ a ɛbɛfata ɛpono biara.Kenkan ne nyinaa → 63, tia $10 a na kyinhyia mua no rehwehwɛ sɛ ɛbɛnya.
Mmuaeɛ (d)
Wɔ $5,000 so: wɔma bɛt sika kɔsi 2,560 kwan; bɛt sika a ɛtɔ so du, 5,120, wɔmpene so. Ahwereɛ nkron bubu no: (19/37)⁹ ≈ 0.25% wɔ kyinhyia mua biara mu, bɛyɛ 11% wɔ agorɔ bere a kyinhyia 100 wom so. Ahwereɛ wɔ mmubuiɛ mu: 10 + 20 + … + 2,560 = $5,110. Agorɔfoɔ no atɔ asiane kɛseɛ a ɛmmaa mma wɔ deɛ ɛyɛ kɛseɛ sen saa boɔ so. Wɔ agorɔ bere pii mu no, mmubuiɛ ka a wɔhwɛ kwan yɛ p × ahwereɛ ≈ 0.0183 × 630 = $11.5 wɔ kyinhyia mua biara mu wɔ $500 ahyeɛ so ne 0.0025 × 5,110 = $12.8 wɔ $5,000 so; sɛ wotwa ano a, ɛyɛ nɔma koro no ara, na ɛyɛ kyinhyia mua no ahwereɛ a wɔhwɛ kwanakontaabuAhwereɛ a wɔhwɛ kwanBoɔ a wɔhwɛ kwan bɔ bɛt sika nyinaa ho; ahwereɛ mfinimfini boɔ wɔ agorɔ bere anaa bere bi mu.Kenkan ne nyinaa → ne $10 a kyinhyia mua a ɛdi nkonim nya no.
Mmuaeɛ (e)
Dabi. Ahwereɛ a wɔhwɛ kwan wɔ kyinhyia mua biara mu = (bɛt sika nyinaa a wɔhwɛ kwan wɔ kyinhyia mua no mu) / 37, wɔ ahyeɛ biara so, ɛfiri sɛ bɛt biara yɛ −1/37 wɔ ne ho so. Ahyeɛ no kyerɛ sɛdeɛ ahwereɛ no kyɛ wɔ nkonim ketewa pii ne ahwereɛ kɛseɛ kakraa bi ntam nko ara. Sɛ ahyeɛ biara nni hɔ na agorɔ dwetiresikaAgorɔ dwetireSika a woayi asie nyinaa ama agorɔ. Agorɔ dwetire so hwɛ ne sɛ wobɛsi gyinaeɛ wɔ dodoɔ a wode bɛto asiane mu wɔ bɛt biara ne agorɔ bere biara mu.Kenkan ne nyinaa → no nni ano a, anka bɛt nsakraeɛ ntoatoasoɔ no bɛ“yɛ adwuma”, wɔ ɔkwan a ɛremmubu da so, na anka ɛbɛto sika a ano nni hɔ nso a n'ahwereɛ a wɔhwɛ kwan yɛ 2.70% wɔ saa sika a ano nni hɔ no so. Ɛnyɛ ɛpono ahyeɛɛponoƐpono ahyeɛBɛt sika ase koraa ne soro koraa a wɔma kwan. Ahyeɛ soro no na ɛsiw mmɔho mmienu bɛt nsakraeɛ ntoatoasoɔ.Kenkan ne nyinaa → no na ɛsi MartingaleɔkwanMartingaleBɛt nsakraeɛ ntoatoasoɔ a ɛbɔ bɛt no mmɔho mmienu ahwereɛ biara akyi. Hwɛ Akwan ɔfa no na hunu deɛ ɛyɛ ne deɛ ɛnyɛ.Kenkan ne nyinaa → kwan sɛ ɛnyɛ adwuma. Akontaabu na ɛsi.
Ɛpono ankasa mmienu
Thorp nhwɛsoɔ yɛ $1 ɛpono a ne ahyeɛ soro yɛ $1,000, a agorɔfoɔ bi ani da so sɛ ɔbɛnya $30 de atɔ anadwo aduane dɔla baako baako. Bɛt sika 1, 2, 4 … 512 ma bɛt du kwan, ɛno nti ahwereɛ du a ɛdi so bubu bɛt nsakraeɛ ntoatoasoɔ no: mpɛn baako wɔ kyinhyia mua 785 mu wɔ Yuropu ntwahonan so (mpɛn baako wɔ 613 mu wɔ Amerika deɛ so), a ne ka yɛ $1,023. Kyinhyia mua aduasa no nyinaa yɛ yie wɔ bere no 96.2% mu. 3.8% a aka no hwere bɛyɛ $1,000, na anadwo no mfinimfini deɛ ɛfiri mu ba yɛ −$9.17, a ɛyɛ $339 a kyinhyia mua aduasa no de to wɔ mfinimfini mu no 1/37 pɛpɛɛpɛ. $5 ɛpono a ne ahyeɛ soro yɛ $2,000 nso nyɛ soronko: bɛt sika kɔsi $1,280 ma bɛt nkron kwan, na mmubuiɛ ba mpɛn baako wɔ kyinhyia mua 403 mu wɔ Yuropu ntwahonan so (mpɛn baako wɔ 323 mu wɔ Amerika deɛ so) na ɛbɔ $2,555 ka.
Agorɔfoɔ a ɔyɛ ɔdefoɔ
Ma bɛt nsakraeɛ ntoatoasoɔ no agorɔ dwetire a emu dɔ sen ahyeɛ a ɛwɔ fam: kyɛfa 1,000, ahyɛaseɛ kyɛfa baako ne kyinhyia 1,000 wɔ kɔkɔɔ so, a wosan hyɛ aseɛ wɔ kyɛfa baako so bere biara a mmɔho mmienu a ɛdi hɔ no bɛsen deɛ aka no. Yɛyɛɛ saa anadwo ɔpepem baako sɔhwɛ. Wɔ Yuropu ntwahonan so no, 52% ba awieeɛ a mfasoɔ wom, 21% hwere biribiara, na mfinimfini deɛ ɛfiri mu ba yɛ kyɛfa −124. Wɔ agorɔ a hweentwahonanHweeTokuru ahabammono baako a sika pɛ bɛt, dumienu kuo bɛt anaa santen tee bɛt biara nkata so; kasino mfasoɔ fibea. Wɔfrɛ no nso hwee baako, 0.Kenkan ne nyinaa → nni mu mpo, a bɛt biara yɛ pɛ no, bɛyɛ 60% pɛ na anka ɛbɛba awieeɛ a mfasoɔ wom. Anadwo a ɛtaa ba yɛ nkonim ketewa; mfinimfini no deɛ, anadwo a ɛba awieeɛ wɔ hwee so no na ɛtua ho ka.
Deɛ yɛsua firi mu
Kasino ahodoɔ de ahyeɛ si hɔ de to wɔn ankasa nsakraeɛakontaabuNsakraeɛNsusuiɛ a ɛkyerɛ sɛdeɛ nsɛm a ɛfiri mu ba no trɛ wɔ boɔ a wɔhwɛ kwan ho. Ɛso ma nɔma baako bɛt, ɛsua ma sika pɛ bɛt.Kenkan ne nyinaa → ano, ɛnyɛ sɛ wɔbɛdi nhyehyɛe so nkonim. Sɛ wodi Martingale so nkonim a, ɛyɛ biribi a ɛka ho a ɛyɛ wɔn dɛ, na anka akontaabu bɛdi so nkonim saa ara. Agorɔfoɔ a ɔsusu sɛ ahyeɛ no ne ɔhaw no ahu abotan ano no na ɔfa no sɛ ɛyɛ siane no.
Akontaabu kwan
Nsɛmmisa
Ɛma wotumi gyina kɔsi ahyeɛ a ɛkɔ soro, baabi a mmubuiɛ no mmaa mma na ɛyɛ kɛseɛ. Ahwereɛ a wɔhwɛ kwan wɔ dɔla biara so nsesa.
Bɛt sika 10, 30, 70, 150, 310: woduru $500 ahyeɛ no so wɔ ahwereɛ enum akyi, na mmubuiɛ no bɔ $570 ka. 2.70% koro no ara.
Sɛdeɛ ɛbɛyɛ a agorɔfoɔ baako rentumi nto sen deɛ kasino no sika a ɛkora bɛtumi atua a ɛnyɛ den. Nsakraeɛ so hwɛ.
Akontaabu ho anigyeɛ
Di kan yɛ eyinom ara wo ho. Wɔakata mmuaeɛ biara so; sɛ woasiesie wo ho a, bue no.
MartingaleɔkwanMartingaleBɛt nsakraeɛ ntoatoasoɔ a ɛbɔ bɛt no mmɔho mmienu ahwereɛ biara akyi. Hwɛ Akwan ɔfa no na hunu deɛ ɛyɛ ne deɛ ɛnyɛ.Kenkan ne nyinaa → bi de $100 hyɛ aseɛ wɔ kɔkɔɔbɛtKɔkɔɔSika pɛ bɛt wɔ nɔma kɔkɔɔ 18 no so. Franse kasa mu: rouge.Kenkan ne nyinaa → so na ɛma ne ho kwan sɛ ɛbɛto bɛt n pɛ wɔ ɔkwan no kyinhyia mua biara mu: ahwereɛ n a ɛtoatoa so akyi no, ɛgyae na ɛsan hyɛ aseɛ. Kyerɛw ɔkwan no kyinhyia mua baako mfinimfini nsunsuansoɔ sɛ formula wɔ n mu, na bu ma bɛt 5, 6 ne 7 wɔ Yuropu ntwahonan so.
Kyerɛ mmuaeɛ no
Mmuaeɛ Mfinimfini boɔ no yɛ 100 × (1 − (38/37)ⁿ): −$14.26 ma bɛt 5, −$17.35 ma 6 ne −$20.52 ma 7. Sɛ woma bɛt pii kwan a, ɛma ɔkwan no kyinhyia mua mfinimfini boɔ yɛ bɔne, ɛnyɛ papa.
Ɔkwan no kyinhyia mua di $100 nkonim gye sɛ bɛt n no nyinaa hwere, a ɛba wɔ hokwan qⁿ mu, baabi a q = 19/37. Sɛ ne nyinaa hwere a, bɛt sika 100, 200 … 100 × 2ⁿ⁻¹ bom yɛ 100 × (2ⁿ − 1). Enti mfinimfini boɔ no yɛ 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Ha no 2q = 38/37, ɛboro 1 kakra, enti (2q)ⁿ nyini bere a n nyini na mfinimfini boɔ no ba fam: n = 5 ma 1.1426 ne −$14.26; n = 6 ma 1.1735 ne −$17.35; n = 7 ma 1.2052 ne −$20.52. Atwedeɛ tenten hwere mpɛn kakra, wɔ ɔkwan no kyinhyia mua 3.6%, 1.8% ne 0.9% mu, nanso ahwereɛ biara kɛseɛ yɛ mmɔho mmienu, na sika pii twa ɛpono no so kyinhyia mua biara, wɔ bɛt biara mu 2.70%.
Wɔ Amerika ntwahonan so no, 2q = 40/38 na nɔma no yɛ −$29.24, −$36.04 ne −$43.20 (Taylor, 2021). Wɔ agorɔ a hweentwahonanHweeTokuru ahabammono baako a sika pɛ bɛt, dumienu kuo bɛt anaa santen tee bɛt biara nkata so; kasino mfasoɔ fibea. Wɔfrɛ no nso hwee baako, 0.Kenkan ne nyinaa → nni mu no, 2q = 1, na ɔkwan no kyinhyia mua biara mfinimfini boɔ yɛ hwee pɛpɛɛpɛ, sɛdeɛ atwedeɛ no tenten te biara.
“Grand Martingale” no ma bɛt sika no yɛ mmɔho mmienu ahwereɛ akyi na ɛde kyɛfasikaKyɛfaBɛt sika a ɛwɔ ase a nhyehyɛe anaa agorɔfoɔ de yɛ adwuma. Wɔka bɛt nsakraeɛ ntoatoasoɔ ho asɛm wɔ kyɛfa mu sɛdeɛ ɛbɛyɛ a ɛbɛfata ɛpono biara.Kenkan ne nyinaa → baako ka ho, enti bɛt sika no kɔ 1, 3, 7, 15, 31 … Woadi ahe so bere a nkonim a edi kan no ba bɛt 1, 2, 3 anaa bɛt a ɛto so k so? Wɔ kyɛfa $10 ne ahyeɛ soro $500 so no, mpɛn ahe na ɔkwan no kyinhyia mua bubu, na ɔkwan no kyinhyia mua gye ahe wɔ mfinimfini mu?
Kyerɛ mmuaeɛ no
Mmuaeɛ Nkonim wɔ bɛt a ɛto so k so ma wodi kyɛfa k so: +1, +2, +3 na ɛkɔ so saa. Wɔ $10 ne $500 so no, ɔkwan no kyinhyia mua bubu ahwereɛ enum akyi, ɔkwan no kyinhyia mua 3.57%, ma −$570. Wɔ mfinimfini mu no, ɔkwan no kyinhyia mua hwere $2.32 wɔ $85.73 a wɔto so mu: 2.70%.
Bɛt sika a ɛto so k yɛ kyɛfa 2ᵏ − 1: 1, 3, 7, 15, 31. Bɛt sika k − 1 a edi kan no bom yɛ 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Nkonim wɔ bɛt a ɛto so k so tua 2ᵏ − 1, enti ɔkwan no kyinhyia mua wie a wadi (2ᵏ − 1) − (2ᵏ − k − 1) = kyɛfa k so: kyɛfa baako foforɔ ma ahwereɛ biara a ɛba ansa na nkonim no aba.
Wɔ $10 kyɛfa biara so no, bɛt sika no yɛ $10, $30, $70, $150 ne $310; deɛ ɛdi so, $630, boro ahyeɛ soro $500 no so. Ahwereɛ enum a ɛtoatoa so, (19/37)⁵ = 3.57%, gye 10 + 30 + 70 + 150 + 310 = $570.
Mfinimfini boɔ: nkonim no de $10 × k ba wɔ hokwan (19/37)ᵏ⁻¹ × 18/37 mu ma k = 1 kɔsi 5, na bubu no gye $570 wɔ hokwan 3.57% mu; ne nyinaa bom −$2.32. Sika a wɔto so wɔ mfinimfini mu yɛ $85.73 ɔkwan no kyinhyia mua biara, na 2.32/85.73 = 2.70%. Martingale kɛkɛ wɔ ɛpono korɔ no ara so bubu ahwereɛ nsia akyi, ɔkwan no kyinhyia mua 1.83%, ma −$630, na ɛhwere $1.74 wɔ $64.20 a wɔto so mu: 2.70% bio.
Agorɔfoɔ bi hyehyɛ Martingale wɔ kɔkɔɔ so a ne bɛt a edi kan yɛ $1 wɔ ɛpono a n'ahyeɛ yɛ $1,000 so, enti bɛt du na ebetumi aba: $1, $2, $4 na ɛkɔ so kɔsi $512. Mpɛn ahe na ɔkwan no kyinhyia mua hwere du no nyinaa, ɛgye bere tenten sɛn ansa na ɛba wɔ mfinimfini mu, na mpɛn ahe na anadwo a bɛt 200 wom wie saa?
Kyerɛ mmuaeɛ no
Mmuaeɛ Ɔkwan no kyinhyia mua hwere du no nyinaa bɛyɛ mprɛ baako wɔ 784 mu, a ɛgye $1,023. Ɛno yɛ bɛyɛ bɛt 1,600 wɔ mfinimfini mu, na anadwo a bɛt 200 wom hyia ɔfasuo no bɛyɛ 11% wɔ bere no mu. Ɔkwan no kyinhyia mua da so hwere $0.31 wɔ mfinimfini mu wɔ $11.31 a wɔto so mu: 2.70%.
Bɛt wɔ kɔkɔɔ so hwere wɔ hokwan 19/37 mu, hwee ka ho. Du a ɛtoatoa so: (19/37)¹⁰ ≈ 0.00128, bɛyɛ 1 wɔ 784 mu. Bɛt sika 1 + 2 + 4 + … + 512 bom yɛ $1,023.
Ɔkwan no kyinhyia mua gye bɛt 2.05 wɔ mfinimfini mu, enti ɔkwan no kyinhyia mua 784 gye bɛyɛ bɛt 1,600. Sɛ wokan no kyinhyia biara a, ahwereɛ du ntoatoasoɔ ba baabi wɔ bɛt 200 mu 11.3% wɔ bere no mu.
Wɔ amanehunu ntam no, agorɔfoɔ no di $1 nkonim ɔkwan no kyinhyia mua biara: bɛyɛ $783 ansa na deɛ ɛyɛ bɔne no aba, afei −$1,023, bɛyɛ −$240 wɔ bere no nyinaa mu. Ɔkwan no kyinhyia mua biara: 1 × (1 − 0.00128) − 1,023 × 0.00128 ≈ −$0.31. Sika a wɔto so wɔ ɔkwan no kyinhyia mua biara mu: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11.31. Na −0.31/11.31 = −2.70%.
Fa toto $10 bɛt a edi kan wɔ ahyeɛ $500 so ho, bɛt nsia: ɔfasuo no ba mprɛ baako wɔ ɔkwan no kyinhyia mua 55 mu, na anadwo a bɛt 200 wom hyia no 84% wɔ bere no mu. Ahyeɛ a ɛkɔ soro ma asɛmmɔne no ba mpɛn kakra na ɛyɛ kɛseɛ, na ɛnyɛ sɛ ne ka yɛ mmerɛ.
Serew kakra
Ɛmmaa mma na ɛyɛ kɛseɛ, wɔ ahyeɛ biara so. Wɔn mmienu a wɔbɔ no nsesa.
Nsɛm fibea
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Thorp, E. O. “Optimal gambling systems for favorable games.”
- Thorp, E. O. The Mathematics of Gambling.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.