Ikibazo: Imbibi z'Ameza
Martingale ya $10 ku meza afite imbibi za $500. Isenyuka kangahe, ihomba angahe iyo isenyutse, kandi ese kuzamura imbibi bifasha?
Ibihombo bitandatu bikurikiranye birayisenya: 1,8% kuri buri cyiciro cy'uburyo, hafi 60% by'imikino y'amazunguruka 100. Isenyuka ritwara $630. Kuzamura imbibi zikagera kuri $5.000 bituma isenyuka riba gake (0,25% kuri buri cyiciro) kandi rikaba rinini ($5.110). Igihombo giteganyijwe kuri buri dolari ryategewe ni 2,70% ku mbibi izo ari zo zose.
Ikibazo
Umukinnyi atega $10 ku mutukuitegaUmutukuItega ryishyura kimwe ku kindi ku mibare 18 y'umutuku. Mu Gifaransa: rouge.Soma byose → kandi agakuba kabiri nyuma ya buri gihombo, agasubira ku ntangiriro nyuma y'intsinzi. Itega rinini ryemewe ku meza ni $500. (a) Ni ibihombo bingahe bikurikiranye bisenya uruhererekane rw'amategaingambaUruhererekane rw'amategaItegeko iryo ari ryo ryose rihindura itega hashingiwe ku bisubizo byabanje. Impererekane mbi ziryongera nyuma yo gutsindwa, inziza nyuma yo gutsinda.Soma byose →? (b) AmahirweimibareAmahirweAmahirwe y'igisubizo, kuva kuri 0 kugeza kuri 1 cyangwa kuva kuri 0% kugeza kuri 100%. Umubare umwe: 1/37 = 2,70%.Soma byose → kuri buri cyiciro ni angahe? (c) Hahombwa iki ku isenyuka? (d) Imbibi za $5.000 zihindura ibisubizo zite? (e) Ese hari imbibi zituma uburyo bwunguka?
Ibisubizo (a)–(c)
Amatega: 10, 20, 40, 80, 160, 320. Irya karindwi ryaba 640 > 500, bityo ibihombo bitandatu birayisenya. (b) p(igihombo) = 19/37 ku mutuku; (19/37)⁶ = 0,0183, hafi 1,8% kuri buri cyiciro. Icyiciro kimara hafi amazunguruka abiri ku mpuzandengo, bityo umukino w'amazunguruka 100 ugira hafi ibyiciro 50 kandi p(nibura isenyuka rimwe) ≈ 1 − 0,9817⁵⁰ ≈ 60%. (c) 10 + 20 + 40 + 80 + 160 + 320 = $630, ari zo initeamafarangaIniteItega ry'ibanze uburyo cyangwa umukinnyi akoresha. Impererekane z'amatega zisobanurwa mu nite kugira ngo zikoreshwe ku meza ayo ari yo yose.Soma byose → z'ibanze 63, ugereranyije na $10 icyiciro cyageragezaga gutsinda.
Igisubizo (d)
Ku $5.000: amatega kugeza kuri 2.560 aremewe; itega rya cumi, 5.120, rirangwa. Ibihombo icyenda birayisenya: (19/37)⁹ ≈ 0,25% kuri buri cyiciro, hafi 11% by'imikino y'amazunguruka 100. Igihombo ku isenyuka: 10 + 20 + … + 2.560 = $5.110. Umukinnyi yaguze ibyago biba gake ku giciro cy'ibyago binini. Mu mikino myinshi, ikiguzi giteganyijwe cy'amasenyuka ni p × igihombo ≈ 0,0183 × 630 = $11,5 kuri buri cyiciro ku mbibi za $500 na 0,0025 × 5.110 = $12,8 kuri $5.000; ni, uretse ukuzengurutsa, umubare umwe, kandi ni igihombo giteganyijweimibareIgihombo giteganyijweAgaciro gateganyijwe gakubye amafaranga yose yategewe; igihombo mpuzandengo mu mukino umwe cyangwa mu gihe runaka.Soma byose → cy'icyiciro wongeyeho $10 ibyiciro byatsinze byinjiza.
Igisubizo (e)
Oya. Igihombo giteganyijwe kuri buri cyiciro = (itega ryose riteganyijwe mu cyiciro) / 37, ku mbibi izo ari zo zose, kuko buri tega ari −1/37 byaryo. Imbibi zigena gusa uko igihombo kigabanywa hagati y'intsinzi nyinshi nto n'ibihombo bike binini. Nta mbibi kandi igishoro cyo gukinaamafarangaIgishoro cyo gukinaAmafaranga yose yashyizwe ku ruhande yo gukina. Gucunga igishoro ni ukwemeza ingano yacyo uzashyira mu kaga kuri buri tega no kuri buri mukino.Soma byose → kitagira iherezo, uruhererekane “rwakora”, mu buryo bw'uko rutazigera rusenyuka, kandi rwanategeka amafaranga atagira iherezo rufite igihombo giteganyijwe cya 2,70% by'ayo mafaranga atagira iherezo. Imbibi z'amezaamezaImbibi z'amezaItega rito n'irinini byemewe. Imbibi zo hejuru ni zo zihagarika impererekane zikuba kabiri.Soma byose → si zo zibuza MartingaleingambaMartingaleUruhererekane rw'amatega rukuba itega kabiri nyuma ya buri gutsindwa. Reba igice cy'Ingamba umenye icyo rukora n'icyo rudakora.Soma byose → gukora. Ni imibare.
Ameza abiri nyayo
Urugero rwa Thorp ni ameza ya $1 afite itega rinini rya $1.000, aho umukinnyi yizeye gutsinda $30 y'ifunguro rya nimugoroba idolari rimwe rimwe. Amatega 1, 2, 4 … 512 yemerera amatega icumi, bityo ibihombo icumi bikurikiranye bisenya uruhererekane: rimwe mu byiciro 785 ku ruziga rw'i Burayi (rimwe muri 613 ku rw'Abanyamerika), ku giciro cya $1.023. Ibyiciro mirongo itatu byose bigenda neza 96,2% by'igihe. Ibindi 3,8% bihomba hafi $1.000, kandi igisubizo cy'impuzandengo cy'umugoroba ni −$9,17, ari yo neza 1/37 cya $339 ibyiciro mirongo itatu bitega ku mpuzandengo. Ameza ya $5 afite itega rinini rya $2.000 ntaho atandukaniye: amatega kugeza kuri $1.280 yemerera amatega icyenda, kandi isenyuka riza rimwe mu byiciro 403 ku ruziga rw'i Burayi (rimwe muri 323 ku rw'Abanyamerika) rigatwara $2.555.
Umukinnyi ukize
Ha uruhererekane igishoro kinini aho kuba imbibi nto: inite 1.000, itega ry'ibanze rya nite imwe n'amazunguruka 1.000 ku mutuku, usubira kuri nite imwe igihe cyose gukuba kabiri gukurikiraho birenze ibisigaye. Twagerageje imigoroba nk'iyo miliyoni imwe. Ku ruziga rw'i Burayi, 52% irangira iri imbere, 21% ihomba byose, kandi igisubizo cy'impuzandengo ni inite −124. No mu mukino utagira zeruuruzigaZeruAkazu kamwe k'icyatsi katafatwa n'itega ryishyura kimwe ku kindi, irya dazeni cyangwa iry'inkingi; ni ho inyungu ya kazino ikomoka. Nanone: zeru imwe, 0.Soma byose →, aho buri tega riboneye, hafi 60% gusa ni yo yarangira iri imbere. Umugoroba usanzwe ni intsinzi nto; impuzandengo yishyurwa n'imigoroba irangirira kuri zeru.
Isomo
Kazino zishyiraho imbibi kugira ngo zigabanye ihindagurikaimibareIhindagurikaIgipimo cy'uko ibisubizo bikwirakwira kure y'agaciro gateganyijwe. Riri hejuru kuri straight up, hasi ku matega yishyura kimwe ku kindi.Soma byose → ryazo bwite, si ukugira ngo zitsinde uburyo bwo gutega. Gutsinda Martingale ni ingaruka y'inyongera zishimira, kandi imibare yari kuyitsinda uko byagenda kose. Umukinnyi utekereza ko imbibi ari zo kibazo yabonye umuhanamanga awitiranya n'umusozi umanuka buhoro.
Formula
Ibibazo
Kigufasha kurokoka kugeza ku mbibi zo hejuru, aho isenyuka riba gake kandi rinini. Igihombo giteganyijwe kuri buri dolari ntigihinduka.
Amatega 10, 30, 70, 150, 310: imbibi za $500 zigerwaho nyuma y'ibihombo bitanu, kandi isenyuka ritwara $570. Ni ya 2,70% nyine.
Kugira ngo umukinnyi umwe adategeka ibirenze ibyo ububiko bwa kazino bushobora kwishyura nta ngorane. Ni ugucunga ihindagurika.
Kwishimisha n'Imibare
Banza ubigerageze ubwawe. Buri gisubizo kirapfunyitse; ugifungure igihe witeguye.
MartingaleingambaMartingaleUruhererekane rw'amatega rukuba itega kabiri nyuma ya buri gutsindwa. Reba igice cy'Ingamba umenye icyo rukora n'icyo rudakora.Soma byose → itangirira kuri $100 ku mutukuitegaUmutukuItega ryishyura kimwe ku kindi ku mibare 18 y'umutuku. Mu Gifaransa: rouge.Soma byose → kandi yiha amatega n menshi cyane muri buri cyiciro: nyuma y'ibihombo n bikurikiranye irareka igatangira bundi bushya. Andika igisubizo cy'impuzandengo cy'icyiciro kimwe nk'inzira y'imibare ishingiye kuri n, maze ukibare ku matega 5, 6 na 7 ku ruziga rw'i Burayi.
Erekana igisubizo
Igisubizo Impuzandengo ni 100 × (1 − (38/37)ⁿ): −$14,26 ku matega 5, −$17,35 kuri 6 na −$20,52 kuri 7. Kwemera amatega menshi bituma icyiciro cy'impuzandengo kiba kibi kurushaho, si cyiza.
Icyiciro gitsinda $100 keretse amatega yose n atsinzwe, ibyo biba ku mahirwe qⁿ, aho q = 19/37. Iyo yose atsinzwe, amatega 100, 200 … 100 × 2ⁿ⁻¹ ateranyijwe aba 100 × (2ⁿ − 1). Bityo impuzandengo ni 100 × (1 − qⁿ) − 100 × (2ⁿ − 1) × qⁿ = 100 × (1 − (2q)ⁿ).
Hano 2q = 38/37, birenga gato 1, bityo (2q)ⁿ ikura uko n ikura kandi impuzandengo ikamanuka: n = 5 itanga 1,1426 na −$14,26; n = 6 itanga 1,1735 na −$17,35; n = 7 itanga 1,2052 na −$20,52. Urwego rurerure ruhomba gake, muri 3,6%, 1,8% na 0,9% by'ibyiciro, ariko buri gihombo kiba kinini incuro ebyiri, kandi amafaranga menshi anyura ku meza muri buri cyiciro, kuri 2,70% bya buri tega.
Ku ruziga rw'Abanyamerika 2q = 40/38 kandi imibare ni −$29,24, −$36,04 na −$43,20 (Taylor, 2021). Mu mukino utagira zeruuruzigaZeruAkazu kamwe k'icyatsi katafatwa n'itega ryishyura kimwe ku kindi, irya dazeni cyangwa iry'inkingi; ni ho inyungu ya kazino ikomoka. Nanone: zeru imwe, 0.Soma byose → 2q = 1, kandi buri cyiciro kigira impuzandengo ya zeru neza uko urwego rwaba rurerure kose.
“Martingale nkuru” ikuba itega kabiri nyuma y'igihombo ikongeraho niteamafarangaIniteItega ry'ibanze uburyo cyangwa umukinnyi akoresha. Impererekane z'amatega zisobanurwa mu nite kugira ngo zikoreshwe ku meza ayo ari yo yose.Soma byose → imwe, bityo amatega aba 1, 3, 7, 15, 31 … Uba uri imbere ho angahe iyo intsinzi ya mbere iza ku itega rya 1, rya 2, rya 3 cyangwa rya k? Kuri nite ya $10 n'itega rinini kurusha rya $500, icyiciro kimeneka kangahe, kandi icyiciro gitwara iki ku mpuzandengo?
Erekana igisubizo
Igisubizo Intsinzi ku itega rya k igusiga uri imbere ho nite k: +1, +2, +3 n'ibindi. Kuri $10 na $500 icyiciro kimeneka nyuma y'ibihombo bitanu, 3,57% by'ibyiciro, kuri −$570. Ku mpuzandengo icyiciro gihomba $2,32 kuri $85,73 zategewe: 2,70%.
Itega rya k ni nite 2ᵏ − 1: 1, 3, 7, 15, 31. Amatega k − 1 ya mbere ateranyijwe aba 2ᵏ − 2 − (k − 1) = 2ᵏ − k − 1. Intsinzi ku itega rya k yishyura 2ᵏ − 1, bityo icyiciro kirangira uri imbere ho (2ᵏ − 1) − (2ᵏ − k − 1) = nite k: nite imwe y'inyongera kuri buri gihombo kibanziriza intsinzi.
Kuri $10 kuri nite amatega ni $10, $30, $70, $150 na $310; irikurikiraho, $630, rirenga itega rinini kurusha rya $500. Ibihombo bitanu bikurikiranye, (19/37)⁵ = 3,57%, bitwara 10 + 30 + 70 + 150 + 310 = $570.
Impuzandengo: intsinzi zinjiza $10 × k ku mahirwe (19/37)ᵏ⁻¹ × 18/37 kuri k = 1 kugeza kuri 5, naho kumeneka gutwara $570 ku mahirwe ya 3,57%; hamwe −$2,32. Amafaranga yategewe agira impuzandengo ya $85,73 kuri buri cyiciro, kandi 2,32/85,73 = 2,70%. Martingale isanzwe ku meza amwe imeneka nyuma y'ibihombo bitandatu, 1,83% by'ibyiciro, kuri −$630, kandi ihomba $1,74 kuri $64,20 zategewe: 2,70% na none.
Umukinnyi ateganya Martingale ku mutuku afite itega rya mbere rya $1 ku meza afite urubibi rwa $1.000, bityo amatega icumi arashoboka: $1, $2, $4 n'ibindi kugeza kuri $512. Icyiciro gitsindwa amatega yose icumi kangahe, ibyo bifata igihe kingana iki ku mpuzandengo, kandi umugoroba w'amatega 200 urangira utyo kangahe?
Erekana igisubizo
Igisubizo Icyiciro gitsindwa amatega yose icumi hafi rimwe muri 784, bitwara $1.023. Ibyo ni hafi amatega 1.600 ku mpuzandengo, kandi umugoroba w'amatega 200 ugonga urukuta hafi 11% by'igihe. Icyiciro kiracyahomba $0,31 ku mpuzandengo kuri $11,31 zategewe: 2,70%.
Itega ku mutuku ritsindwa ku mahirwe ya 19/37, zeru ibariwemo. Icumi bikurikiranye: (19/37)¹⁰ ≈ 0,00128, hafi 1 muri 784. Amatega 1 + 2 + 4 + … + 512 ateranyijwe aba $1.023.
Icyiciro gifata amatega 2,05 ku mpuzandengo, bityo ibyiciro 784 bifata hafi amatega 1.600. Ubaze izunguruka ku rindi, uruhererekane rw'ibihombo icumi ruboneka ahantu runaka mu matega 200 11,3% by'igihe.
Hagati y'ibyago umukinnyi atsinda $1 kuri buri cyiciro: hafi $783 mbere y'icyago, hanyuma −$1.023, hafi −$240 mu gihe cyose. Kuri buri cyiciro: 1 × (1 − 0,00128) − 1.023 × 0,00128 ≈ −$0,31. Amafaranga yategewe kuri buri cyiciro: 1 + 2 × (19/37) + 4 × (19/37)² + … + 512 × (19/37)⁹ ≈ $11,31. Kandi −0,31/11,31 = −2,70%.
Gereranya n'itega rya mbere rya $10 ku rubibi rwa $500, amatega atandatu: urukuta ruza rimwe mu byiciro 55, kandi umugoroba w'amatega 200 uruhura na rwo 84% by'igihe. Urubibi rwo hejuru rutuma isenyuka riba gake kandi rinini kurushaho, si irihendutse.
Akanya k'urwenya
Gake kandi binini, ku mbibi izo ari zo zose. Igikubo ntigihinduka.
Inkomoko
- 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.