Kusantowa a dogon zango: dokar manyan lambobi a roulette
Yadda matsakaicin sakamako ke daidaita zuwa −2.70%, da kuma dalilin da ya sa hakan ke ɗaukar lokaci fiye da yadda mutum ke zato.

Matsakaicin asara a kowane juyi yana nufar −1/37 yayin da juye-juye ke ƙaruwa, kuma bazuwar sakamakon tana raguwa kamar 1/√N. Bayan juye-juye 1,000 na caca a kan ja, yawanci matsakaicin yana cikin ±3% na abin da ake tsammani.
Dokar manyan lambobi
Matsakaicin sakamakon cace-cace masu zaman kansu yana nufar ƙimar da ake tsammanilissafiƘimar da ake tsammaniMatsakaicin sakamako a kowace caca a cikin dogon lokaci. −1/37 na kuɗin caca a kowace cacar Turai ta yau da kullum.Karanta gaba ɗaya →. Bayan juye-juye N na cacar ɗaya zuwa ɗaya da raka'akuɗiRaka'aKuɗin caca na tushe da tsari ko ɗan wasa ke aiki da shi. Ana bayyana jerin ƙarin caca a raka'a don su yi aiki a kowane tebur.Karanta gaba ɗaya → ɗaya, matsakaicin sakamako a kowane juyi yana da abin da ake tsammani na −1/37 da standard deviationlissafiStandard deviationTushen murabba'i na bambancin sakamako; nisan da sakamako ke yawan yi daga ƙimar da ake tsammani.Karanta gaba ɗaya → (ma'aunin bazuwa) na 1/√N. A N = 100, wannan bazuwar ta kai 10% na kuɗin caca, kusan sau huɗu na alamar 2.7%. A N = 1,000 ta kai 3.2%, kusan daidai da alamar. A N = 10,000 ta kai 1%, kuma fifikon gidan caca ya bayyana ƙarara. Fifikon ba ya ganuwa a zaman wasa ɗaya, amma ba za a iya musanta shi a cikin shekara guda ba.
Abin da ke daidaita da abin da ba ya daidaita
Matsakaici a kowane juyi yana daidaita. Jimilla ba ta daidaita: tana ƙara nisa daga sifili yayin da N ke girma, da kusan 2.7% na N a abin da ake tsammani, tare da bazuwar √N a kewayensa. Ana tsammanin ɗan wasan da ya buga juye-juye 10,000 da $10 ya yi asarar $2,700, ƙari ko ragi $1,000. Kason jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → a cikin sakamakon ya daɗe da daidaita kusa da 48.65%; amma ragowar kuɗin ɗan wasa bai daidaita a ko'ina ba.
- p10 −$440
- p25 −$280
- tsakiya −$140
- p75 +$0
- p90 +$140
Dalilin da ya sa tunanin zuci ke kuskure
Mutane suna tsammanin fifikon zai bayyana da sauri saboda 2.70% yana jin kamar mai yawa, kuma suna tsammanin dogon wasa zai “daidaita” saboda dokar manyan lambobilissafiDokar manyan lambobiMatsakaicin sakamako masu zaman kansu yana kusantar ƙimar da ake tsammani yayin da yawan gwaje-gwaje ke ƙaruwa.Karanta gaba ɗaya → ta ce matsakaici yana daidaita. Duk tunanin biyu suna kuskuren girman abin: fifikon ƙarami ne idan aka kwatanta da bambancin sakamakolissafiBambancin sakamakoMa'aunin yadda sakamako ke yaɗuwa a kewayen ƙimar da ake tsammani. Yana da girma ga straight up, ƙanƙanta ga cace-cacen ɗaya zuwa ɗaya.Karanta gaba ɗaya → na juyi ɗaya (σ = raka'a 1 a kan EV = 0.027), don haka yana ɗaukar (1/0.027)² ≈ juye-juye 1,400 kafin ya bayyana a matakin sigma ɗaya. Kuma daidaitar matsakaici ba ta jawo jimilla ta dawo baya; tana kawai rage nauyin sauye-sauyen da suka gabata.
Daidaitawa a sauran cace-cace
Ga cacar straight upcacaStraight upLamba ɗaya tak, biya 35:1.Karanta gaba ɗaya → (cacar lamba ɗaya) σ = 5.84, don haka lissafi iri ɗaya yana ba da (5.84/0.027)² ≈ juye-juye 47,000 kafin fifikon ya kai sigma ɗaya. Ɗan wasan straight up zai iya yin wasa na shekaru ba tare da an iya bambanta fifikon da sa'a a cikin bayanansa ba, alhali yana biyan 2.70% daidai da ɗan wasan ja wanda ke ganinsa cikin ʼyan makonni.
Abin da hakan ke nufi ga gwajin tsare-tsare
Duk wanda ya ce wani tsari yana aiki ya kamata ya iya nuna shi a kan juye-juye 10,000 ko fiye tare da matsakaici mai riba da aka rubuta. A juye-juye 1,000, matsakaici mai riba a cacar ɗaya zuwa ɗaya abu ne da ke faruwa sau ɗaya cikin biyar ko da tsarin ba shi da wani tasiri; a juye-juye 100 kusan kamar jefa tsabar kuɗi ne. Simulators (na'urorin kwaikwayo) na wannan shafin suna gudanar da zaman wasa 2,000 na tsawon da kuka zaɓa a cikin daƙiƙa ɗaya, wanda shaida ce fiye da kowace shaidar baki.
Ƙa'idar lissafi
Tambayoyi
YiwuwarlissafiYiwuwaDamar faruwar wani sakamako, daga 0 zuwa 1 ko daga 0% zuwa 100%. Lamba ɗaya: 1/37 = 2.70%.Karanta gaba ɗaya → sake samun riba tana raguwa zuwa sifili yayin da kuke wasa. Jimilla tana ci gaba da sauka ba tare da iyaka ba.
A'a. Tana nufin sauye-sauyen da suka gabata suna zama ƙaramin kaso na babban samfuri.
Ga cacar ɗaya zuwa ɗaya, kusan 10,000 kafin fifikon ya bayyana ƙarara. Ga straight up, kusan 50,000.
Nishaɗi da lissafi
Ku fara gwada waɗannan da kanku. Kowace mafita a naɗe take; ku buɗe ta idan kun shirya.
Abokai biyu suna buga zagaye 20 na wasa mai adalcilissafiWasa mai adalciWasan da babu ɓangaren da ke da fifiko: kowace caca tana da ƙimar da ake tsammani ta sifili.Karanta gaba ɗaya →, jefa kwabo ko jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → a kan baƙicacaBaƙiCacar ɗaya zuwa ɗaya (even-money) a kan lambobi baƙi 18. A Faransanci: noir.Karanta gaba ɗaya → inda ake mayar da caca idan 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 → ya fito, kuma suna riƙe maki mai gudana. Ku ƙirga zagayen da kowannensu ke a gaba (canjaras yana ƙirguwa ga wanda ya kasance a gaba kafin hakan). Wanne ya fi yiwuwa: a raba jagorancin zagaye 10 da 10, ko aboki ɗaya ya jagoranci duka 20?
Nuna mafita
Amsa Aboki ɗaya ya jagoranci duka zagaye 20 ya fi yiwuwa nesa ba kusa ba: 35.2%, idan aka ƙirga kowane ɗaya daga cikin abokan, idan aka kwatanta da 6.1% ga rabo daidai na 10 da 10.
Damar da maki mai gudana na adalci zai kasance daidai bayan zagaye 2k ita ce u = C(2k, k)/2²ᵏ: 0.246 bayan zagaye 10 da 0.176 bayan 20. Wani sanannen sakamako, dokar arc-sine, ya ce damar da wani aboki da aka ambata zai jagoranci daidai zagaye 2k cikin 2n ita ce u(2k) × u(2n − 2k).
Jagorancin duka 20: u(20) × u(0) = 0.176 ga kowane aboki, 35.2% ga kowane ɗayansu. Rabo daidai: u(10) × u(10) = 0.246² = 6.1%, rabon da ya fi rashin yiwuwa duka.
Don haka a wasa mai adalci jagoranci ba kasafai yake canja hannu ba. Ɗan wasan da ya kasance a gaba duk maraice, ko a baya duk maraice, bai ga wani abu na ban mamaki ba, kuma ba ya nuna komai game da dabaran ko game da gwaninta. Epstein (2013) ya ba da doka iri ɗaya ga wasanni masu tsawo: rabin lokaci ɓangaren da ya fi sa'a yana jagoranci fiye da kashi 85% na wasan.
Ɗan barkwanci

Matsakaici yana daidaita. Jimilla kuwa tana tafiya kawai.
Majiyoyi
- Bernoulli, J. Ars Conjectandi.
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.