Yiwuwar jerin sakamako iri ɗaya a roulette
Dogayen jerin launi ɗaya abu ne na yau da kullum. Kuma ba su da wata ma'ana ga juyi na gaba.
Ja sau biyar a jere: 2.7%. Sau goma: 0.074%. A zaman wasa na juyi 200, jerin launi ɗaya sau shida yana fitowa fiye da rabin lokaci.
Sau nawa jerin asara ke zuwa
Cacar ɗaya zuwa ɗaya a kan dabaran mai sifili ɗaya tana yin asara a juyi 19 cikin 37. Kowane layi yana nuna yiwuwar cewa wasu juye-juye a jere duk za su yi asara, da abin da ɗan wasan Martingale, wanda ke ninka caca bayan kowace asara, ya rasa zuwa lokacin.
| asara a jere | yiwuwa | kusan 1 cikin | asara zuwa yanzu | caca ta gaba |
|---|---|---|---|---|
| 1 | 51.4% | 2 | $1 | $2 |
| 2 | 26.4% | 4 | $3 | $4 |
| 3 | 13.5% | 7 | $7 | $8 |
| 4 | 7.0% | 14 | $15 | $16 |
| 5 | 3.6% | 28 | $31 | $32 |
| 6 | 1.8% | 55 | $63 | $64 |
| 7 | 0.94% | 106 | $127 | $128 |
| 8 | 0.48% | 207 | $255 | $256 |
| 9 | 0.25% | 403 | $511 | $512 |
| 10 | 0.13% | 784 | $1,023 | $1,024 |
| 11 | 0.065% | 1527 | $2,047 | $2,048 |
| 12 | 0.034% | 2974 | $4,095 | $4,096 |
| 13 | 0.017% | 5792 | $8,191 | $8,192 |
| 14 | 0.0089% | 11279 | $16,383 | $16,384 |
| 15 | 0.0046% | 21965 | $32,767 | $32,768 |
Yiwuwar ita ce 19/37 da aka ninka da kanta sau ɗaya a kowace asara, ana ƙirgawa daga juyi ɗaya ƙayyadadde. Zaman wasa mai tsawo yana da juye-juye da yawa inda irin wannan jeri zai iya farawa, don haka a cikin yamma guda ya fi yiwuwa nesa ba kusa ba fiye da yadda layi ɗaya ke nunawa.
Jeri ɗaya
YiwuwarlissafiYiwuwaDamar faruwar wani sakamako, daga 0 zuwa 1 ko daga 0% zuwa 100%. Lamba ɗaya: 1/37 = 2.70%.Karanta gaba ɗaya → cewa juye-juye k na gaba duka jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → ne ita ce (18/37)ᵏ. Biyu: 23.7%. Biyar: 2.7%. Goma: 0.074%, kusan 1 cikin 1,350. Ashirin: 1 cikin miliyan 1.8. Ashirin da shida, jeri mafi tsawo a tarihin Monte Carlo: 1 cikin miliyan 137. Waɗannan su ne yiwuwar wani jeri na musamman da zai fara a wani juyi na musamman, kuma wannan tambaya ce da kusan babu wanda ke yin ta a zahiri.
A cikin zaman wasa
Tambayar da mutane ke nufi ita ce “shin jeri na k zai bayyana a wani wuri a zaman wasana?”, kuma amsar ta fi (18/37)ᵏ girma ƙwarai, domin zaman wasa na juyi 200 yana ba da wurare kusan 200 da jeri zai iya farawa. Jerin launi iri ɗaya sau shida ko fiye, ja ko baƙicacaBaƙiCacar ɗaya zuwa ɗaya (even-money) a kan lambobi baƙi 18. A Faransanci: noir.Karanta gaba ɗaya →, yana bayyana a zaman wasa na juyi 200 kusan 94% na lokaci; sau takwas ko fiye kusan 47%; sau goma ko fiye kusan 14%. Idan aka faɗi launin tun kafin a fara, damar tana raguwa, zuwa kusan 75%, 27% da 7%. Ɗan wasan da ya zauna dare guda zai ga dogayen jerin sakamako iri ɗaya a kowane dare.
Me ya sa wannan ke da muhimmanci ga Martingale
Kowane jerin ƙarin cacadabaraJerin ƙarin cacaDuk wata ƙa'ida da ke canza kuɗin caca bisa sakamakon da ya gabata. Jerin ƙarin caca bayan asara yana ɗaga shi bayan an rasa, na bayan nasara kuma bayan an ci.Karanta gaba ɗaya → bayan asara yana da tsawon jerin da ke karya shi: asara shida daga tushe na $10 a iyakar $500, goma ga FibonaccidabaraFibonacciJerin ƙarin caca da ke bin jerin lambobi 1, 1, 2, 3, 5, 8, 13…: mataki ɗaya gaba bayan asara, matakai biyu baya bayan nasara.Karanta gaba ɗaya →, biyar ga ninka sau uku a dozincacaDozin1–12, 13–24 ko 25–36, biya 2:1.Karanta gaba ɗaya →. Yiwuwar jerin asara sau shida a cacar ɗaya zuwa ɗaya ita ce (19/37)⁶ ≈ 1.8% a kowane ƙoƙari, kuma zaman wasa na juyi 200 yana ƙunshe da wataƙila ƙoƙari 100, don haka karyewar tana zuwa a kusan zaman wasa huɗu cikin biyar na irin wannan tsawo. Yiwuwar jerin sakamako iri ɗaya a roulette ita ce dalilin da ya sa wannan faɗuwar ba abu ne da ba kasafai ake gani ba.
Lambobi masu zafi da masu sanyi
Lissafi iri ɗaya yana aiki ga straight upcacaStraight upLamba ɗaya tak, biya 35:1.Karanta gaba ɗaya →. Cikin juyi 300, adadin da ake tsammani ga kowace lamba 8.1 ne; standard deviationlissafiStandard deviationTushen murabba'i na bambancin sakamako; nisan da sakamako ke yawan yi daga ƙimar da ake tsammani.Karanta gaba ɗaya → kusan 2.8 ne. Wata lamba za ta nuna nasara 15, wata kuma 2, ta hanyar kwatsam kawai, a kowane lokaci. Allon sakamakon da ke haskaka lambobi “masu zafi” yana haskaka saman normal distribution ne. Ɗakin gwajin roulette yana bayyana yawan juye-juyen da ake buƙata kafin lamba mai zafitatsuniyaLamba mai zafiLambar da ta fito fiye da yadda ake tsammani a kwanan nan. A dabaran da ke da adalci ba ta hasashen komai.Karanta gaba ɗaya → ta sami wata ma'ana (dubbai, daga dabaran ɗaya da aka sani).
Rashin dogaro da juna shi ne babban batu
Jerin ja sau goma ba ya sa baƙi ya fi yiwuwa, kuma ba ya sa ja ya fi yiwuwa. Duka biyun 18/37 ne. Jerin sakamako iri ɗaya siffa ce ta jerin sakamako na kwatsam, kuma rashinsu ne zai zama abin zargi: jerin juyi 200 da babu jeri da ya wuce huɗu zai zama shaida mafi ƙarfi ta magudi fiye da wanda ke da jeri na goma sha biyu.
Ƙa'idar lissafi
Tambayoyi
Baƙi 26 a Monte Carlo a ranar 18 ga Agusta, 1913 shi ne mafi shahara. An ba da rahoton ja 32 a jere a wani gidan caca na Amurka a 1943. Duka biyun sun dace da kwatsam idan aka yi la'akari da yawan juye-juyen da gidajen caca ke yi.
18/37 ne ko ta wace hanya. Ku yi cacar da shirinku ya ce, ko kada ku yi komai.
A'a. Karkata tana bayyana ne yayin da wasu ramukakayan aikiRamiƊaya daga cikin ɗakuna 37 masu lamba a kan rotor. Ana kuma kiransa: slot, cup.Karanta gaba ɗaya → ke fitowa fiye da sauran cikin dubban juye-juye, ba a matsayin jerin launi ba.
Nishaɗi da lissafi
Ku fara gwada waɗannan da kanku. Kowace mafita a naɗe take; ku buɗe ta idan kun shirya.
Juye-juye nawa na dabaran roulette na Turai ake buƙata, a matsakaici, kafin kowace ɗaya daga cikin lambobi 37 ta fito aƙalla sau ɗaya?
Nuna mafita
Amsa Kusan juye-juye 155 a matsakaici; tsaka-tsakin (median) shi ne 147.
Ku ƙirga jiran sabuwar lamba ɗaya bayan ɗaya. Juyi na farko koyaushe yana nuna sabuwar lamba. Bayan kun ga lamba 1, ramukakayan aikiRamiƊaya daga cikin ɗakuna 37 masu lamba a kan rotor. Ana kuma kiransa: slot, cup.Karanta gaba ɗaya → 36 cikin 37 sababbi ne, don haka sabuwar lamba ta gaba tana ɗaukar juye-juye 37/36 a matsakaici. Bayan kun ga lambobi k, tana ɗaukar 37/(37 − k).
Ku haɗa jiran: 37/37 + 37/36 + 37/35 + … + 37/1 = 37 × (1 + 1/2 + 1/3 + … + 1/37) ≈ 37 × 4.20 = juye-juye 155.5.
Yawancin wannan yana ƙarshe. Lambar ƙarshe ita kaɗai tana ɗaukar juye-juye 37 a matsakaici, biyar na ƙarshe kuma kusan 85. A dabaran roulette na Amurka, irin wannan jimlar tana ba da kusan 161. Wannan shi ne matsalar mai tattara takardun coupon (coupon collector’s problem); matsakaicin iri ɗaya yana fitowa daga lissafin da aka saba.
A cikin juye-juye 37 na dabaran roulette na Turai, juyi ɗaya ga kowane rami, lambobi daban-daban nawa ya kamata ku yi tsammanin gani?
Nuna mafita
Amsa Kusan 23.6. Kusan sulusin lambobin, wajen 13, ba sa fitowa ko kaɗan.
Ku ɗauki lamba ɗaya, misali 17. A kowane juyi damar rashin fitowarta 36/37 ce, don haka damar ta ɓace a duka juye-juye 37 ita ce (36/37)³⁷ ≈ 0.363.
Kowace lamba tana da dama iri ɗaya, don haka a matsakaici lambobi 37 × 0.363 ≈ 13.4 ba sa fitowa, 37 − 13.4 ≈ 23.6 kuma suna fitowa, da yawa daga cikinsu sau biyu ko fiye.
ʼYan wasan roulette suna kiran wannan “dokar sulusi” (law of thirds). Ba doka ba ce, lissafin maimaitawa ne kawai: a cikin jerin da tsawonsa ya kai na dabaran, kusan sulusin lambobin ba sa fitowa, a dabaran mai adalci kamar a kowane. A kan ramuka 38 kason shi ne (37/38)³⁸ ≈ 0.363 shi ma; ga babban dabaran yana tsayawa kusa da 1/e ≈ 0.368.
Wani abokinku ya nuna muku jerin launuka biyu na 50, jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → ko baƙicacaBaƙiCacar ɗaya zuwa ɗaya (even-money) a kan lambobi baƙi 18. A Faransanci: noir.Karanta gaba ɗaya →, kowanne an ce ya fito ne daga dabaran da aka cire sifilidabaranSifiliRamin kore guda ɗaya da babu cacar ɗaya zuwa ɗaya, ta dozin ko ta ginshiƙi da ke rufe shi; tushen fifikon gidan caca. Ana kuma kiransa: sifili ɗaya, 0.Karanta gaba ɗaya →. Ɗaya yana sauya launi sau 25 kuma jerinsa mafi tsawo na shida ne. Ɗayan yana sauyawa sau 35 kuma bai taɓa wuce uku a jere ba. Wanne aka ƙirƙira?
Nuna mafita
Amsa Na biyun. Jerin gaske yana sauyawa kusan sau 24 ko 25. Jerin da ke da sauyi 35 yana fitowa kusan sau ɗaya cikin jeri 500, wanda kuma ba shi da jerin da ya wuce uku kusan sau biyu cikin 100.
Kowane sabon launi ko dai yana maimaita na baya ko yana sauyawa, rabi da rabi, don haka damar sauyi 49 tana ba da sauyi 24.5 a matsakaici, da ƙari ko ragi na kusan 3.5. Talatin da biyar ya wuce matsakaicin da irin waɗannan matakai uku; dabaran mai adalci yana samar da sauyi da yawa haka kusan sau ɗaya cikin jeri 500.
Jerin sakamako iri ɗaya yana ba da labari iri ɗaya. Jerin gaske na 50 yana da jerin huɗu ko fiye kusan kashi 98% na lokaci, kuma jerinsa mafi tsawo galibi na biyar ko shida ne.
Mutanen da aka nemi su ƙirƙiri jerin kwatsam suna sauyawa da yawa, domin jerin huɗu ko biyar yana yi musu kamar tsari. Taylor (2021) ya nuna wannan alamar a cikin jerin jefa kwabo da ɗalibai suka ƙirƙira: kwatsam na gaske yana da jerin sakamako iri ɗaya fiye da yadda mutane ke tsammani.
Wani littafin labaran gidan caca ya ba da labarin wani ɗan wasa da lambarsa ta fito sau biyar a jere, ya kuma kira shi tarihi (record). Yaya wuyar hakan yake a dabaran roulette na Turai? Kuma sau nawa ya kamata ya faru a wani wuri a gidan caca ɗaya mai teburai 20, kowanne yana juyawa sau 40 a awa, dare da rana?
Nuna mafita
Amsa Lambar da aka ambata ta fito sau biyar a jere: 1 cikin miliyan 69. Kowace lamba ta sake fitowa sau huɗu bayan ta fara fitowa: 1 cikin miliyan 1.87. Wannan gidan caca yana juyawa kusan sau miliyan 7 a shekara, don haka ya kamata ya ga jerin biyar sau uku ko huɗu a shekara.
Lambar da aka ambata tana fitowa da damar 1/37 a kowane juyi, don haka biyar a jere suna da damar (1/37)⁵ = 1 cikin 69,343,957. Amma babu wanda ke ambaton lambar tun da farko. Duk abin da ya fara fitowa, juye-juye huɗu na gaba dole su maimaita shi: (1/37)⁴ = 1 cikin 1,874,161.
Teburai 20 × juye-juye 40 × awa 24 × kwana 365 = juye-juye 7,008,000 a shekara. Kowane juyi zai iya fara jeri, don haka gidan cacar yana tsammanin 7,008,000/1,874,161 ≈ jerin biyar 3.7 a shekara.
Babu wani tarihi na hukuma irin wannan, kuma ba a buƙatarsa: abin da ba kasafai yake faruwa a tebur ɗaya ba abu ne na yau da kullum a faɗin gidan caca, kuma ana ba da labarin teburin da abin ya faru. Labarin, a Goldman (2007), ya faru ne a dabaran mai sifili biyu, inda alƙaluman su ne 1 cikin miliyan 79 da 1 cikin miliyan 2.09.
Ɗan barkwanci
Dogayen jeri su ne kamannin kwatsam.
Majiyoyi
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed.