Kuskuren ɗan caca
Imanin cewa “lokacin” wani sakamako ya yi saboda bai fito ba na ɗan lokaci.

Bayan ja sau goma, yiwuwar baƙi har yanzu 18/37 ce. Dabaran ba ya ajiye wani tarihi kuma ba shi da bashin kowa.
Kuskuren
Idan jacacaJaCacar ɗaya zuwa ɗaya (even-money) a kan lambobi ja 18. A Faransanci: rouge.Karanta gaba ɗaya → ya fito sau goma, ʼyan wasa da yawa suna jin cewa “lokacin” baƙicacaBaƙiCacar ɗaya zuwa ɗaya (even-money) a kan lambobi baƙi 18. A Faransanci: noir.Karanta gaba ɗaya → ya yi, kuma suna yin caca bisa haka. Wasu kuma suna jin cewa ja “yana da zafi” kuma suna yin caca a kansa. Dukansu suna tunani ne daga jerin sakamakon da ya riga ya ƙare zuwa juyin da bai faru ba tukuna. Ƙwallokayan aikiƘwalloƘaramin ƙwallon da ake jefawa a hanyarsa a akasin juyawar rotor. Ƙwallayen zamani na ivorine ko Teflon ne, 18–21 mm.Karanta gaba ɗaya → bai san abin da juye-juye goma na ƙarshe suka kasance ba; ramukakayan aikiRamiƊaya daga cikin ɗakuna 37 masu lamba a kan rotor. Ana kuma kiransa: slot, cup.Karanta gaba ɗaya → girmansu ɗaya ne da yadda suke sa'a ɗaya da ta wuce; yiwuwarlissafiYiwuwaDamar faruwar wani sakamako, daga 0 zuwa 1 ko daga 0% zuwa 100%. Lamba ɗaya: 1/37 = 2.70%.Karanta gaba ɗaya → baƙi 18/37 ce kamar yadda take koyaushe.
Me ya sa yake kama da gaskiya
A dogon lokaci, kason ja da baƙi lallai suna kusantar 18/37 kowanne. 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 → tana tabbatar da haka. Amma wannan kusantar tana faruwa ne ta hanyar narkewa, ba ta hanyar gyara ba: bayan ƙarin juyi 1,000, ƙarin ja goma ɗin sun zama ƙaramin abu da ba a iya gani a kason, ba don baƙi ya kamo ba, sai don samfurin ya girma. Babu wata hanyar da ke sa baƙi ya fi yiwuwa, kuma dokar manyan lambobi ba ta buƙatar irin wannan hanya.
Monte Carlo, 18 ga Agusta, 1913
Baƙi ya fito sau 26 a jere a wani teburi a Casino de Monte-Carlo. Tun daga kusan baƙi na goma sha biyar, ʼyan wasa suka yi ta tururuwa zuwa ja, suna ninkawa suna sake ninkawa, kowane juyi “ya fi tabbas” fiye da na baya. Gidan caca ya ci miliyoyin francs. Jerin yana da yiwuwar 1 cikin miliyan 137 tun daga baƙi na farko, kuma yiwuwar cewa zai faru a ƙarshe a wani wuri a gidajen caca na duniya kusan ɗaya ce. Shi ya sa ake kuma kiran kuskuren ɗan caca da kuskuren Monte Carlo.
Ba abin mamaki ba. Yana faruwa sau ɗaya cikin 44.
ʼYan wasa sun fara lura. “Lokacin” ja ya yi.
Teburin ya cika maƙil. Ana ninka kuɗin caca a kan ja bayan kowace asara.
An yi asarar dukiya. Kowane sabon baƙi yana sa ja ya zama kamar ya fi tabbas.
Jerin ya ƙare. Gidan caca ya ci miliyoyin francs daga mutanen da ke caca a kan jerin da ba zai iya “sanin” cewa jeri ne ba.
Hannu mai zafi
Bin launi saboda yana cikin jeri kuskure ɗaya ne da aka juya: ana tsammanin jerin zai ci gaba maimakon ya ƙare. Masana halayyar ɗan adam suna kiran sa hannu mai zafitatsuniyaHannu mai zafiImanin cewa jerin nasarori, ko na sakamako ɗaya, yana sa a fi yiwuwar sake samun irinsa. A kan dabaran, kuskuren ɗan caca ne aka juya shi.Karanta gaba ɗaya →, bayan wani bincike na 1985 da Gilovich, Vallone da Tversky suka yi, wanda ya nemi jerin nasara a jefa ƙwallon kwando, amma bai sami komai fiye da kwatsam ba. A 2018, Miller da Sanjurjo sun nuna cewa gwajin kansa ya karkata: a cikin gajeren rikodi, kason nasara nan da nan bayan nasara yana fitowa ƙasa da kason nasarar ɗan wasan gaba ɗaya, a matsakaici, don haka ɗan wasan da ba shi da hannu mai zafi ko kaɗan yana kama da ɗan sanyi. Bayan gyara, bayanan iri ɗaya suna nuna ɗan ƙaramin hannu mai zafi. Mutum zai iya ƙara kuzari; dabaran kuwa ba shi da wani yanayi na zafi ko sanyi. Wasu ʼyan wasa suna riƙe imani biyun a lokaci ɗaya, cewa jeri yana ci gaba har sai ya yi “tsawo da yawa” sannan ya ƙare. A dabaran adalci, kowane nau'i yana da 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 → iri ɗaya da zaɓar launi ta hanyar jefa kwabo: −2.70% na kuɗin caca.
A ainihin teburi
Croson da Sundali (2005) sun kalli bidiyon tsaro na sa'o'i 18 daga teburi ɗaya mai 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 → biyu a wani babban gidan caca a Reno: ʼyan wasa 139 da suka yi cace-cace 24,131. A cace-cacen ɗaya zuwa ɗayacacaCacar ɗaya zuwa ɗaya (even-money)Ja/baƙi, mara/cika, manya/ƙanana. Lambobi 18, biya 1:1.Karanta gaba ɗaya →, ʼyan wasa sun rabu kusan daidai tsakanin masu bin gajerun jeri da masu adawa da su. Bayan sakamako iri ɗaya sau biyar a jere, sun karkata zuwa adawa da jerin, kuma bayan sau shida ko fiye, 85% na cace-cacensu suna gefe ɗayan. Su kansu ʼyan wasan nan sun nuna hannu mai zafi game da kansu: bayan nasara a lambobi, sun rufe lambobi 13.6 a juyi na gaba a matsakaici, idan aka kwatanta da 9.2 bayan asara. Dabaran nasu bai nuna wata alamar karkata ba, don haka babu ɗaya daga cikin waɗannan da ya canza abin da kowannensu zai iya tsammani.
A wajen gidan caca
Irin wannan jan hankali yana bayyana a inda ake yanke shawarwari na gaske. Clotfelter da Cook (1993) sun bi wasan lambobi na yau da kullum na Maryland: kuɗin da aka saka a kan wata lamba yana raguwa ƙwarai da zarar an zana ta, kuma yana ɗaukar watanni kafin ya farfaɗo, ko da yake zanen ba shi da ƙwaƙwalwa. Chen, Moskowitz da Shue (2016) sun same shi a cikin ƙwararru. Alƙalan neman mafaka na Amurka sun fi yiwuwar ƙin ƙara da har zuwa maki 3.3 na kashi (percentage points) idan sun amince da wadda ta gabace ta. Yiwuwar alƙalan wasan baseball su kira strike tana raguwa da maki 1.5 bayan sun kira ɗaya, kuma fiye da ninkin haka a jefe-jefen da ke da wuyar tantancewa. A wani gwaji na fili da jami'an bashi da ake biya kuɗi ƙayyadadde, an juya har zuwa 9% na shawarwari ta wannan hanya. Ƙararrakin sun zo ba bisa wani tsari ba, don haka shawarar ƙarshe ba ta faɗi komai game da ta gaba ba, kamar yadda juyi na ƙarshe ba ya faɗin komai game da na gaba.

Kuskuren ɗan caca na akasi
Masanin falsafa Ian Hacking (1987) ya ba da suna ga nau'i na uku. Wani ya shiga ɗaki, ya ga wasu dice biyu sun faɗi shida-shida, sai ya ɗauka cewa tabbas an jefa su sau da yawa a baya. Kowane jefi yana da yiwuwa iri ɗaya na zama shida-shida, na farko na daren ko na dubu, don haka ganin hakan ba ya faɗin komai game da tsawon lokacin da wasan ya ɗauka. A roulette, shi ne ɗan wasan da ya iso ya ga allo da ke nuna ja takwas, sai ya ɗauka cewa tabbas teburin ya yi ta juyawa tsawon sa'o'i kafin ya samar da su. Kuskuren na yau da kullum yana ɗora abin da ya wuce a kan juyi na gaba; wannan kuma yana danganta wani abu da ba kasafai ake gani ba da wani abin da ya wuce wanda babu wanda ya gani.
Masanin halayyar ɗan adam da ya yarda da shi
Kuskuren ya sami masu kare shi daga cikin masana. Masanin halayyar ɗan adam ɗan ƙasar Jamus, Karl Marbe, ya duba rajistan haihuwa huɗu masu haihuwa 49,152 kowanne, ya gano cewa jerin maza ko na mata mafi tsawo a jere 17 ne, kuma ya yanke hukunci cewa dogayen jeri sun fi ƙaranci fiye da yadda kwatsam ke bari, don haka jeri yana sa ɗayan sakamakon ya fi yiwuwa. Kwatsam kaɗai yana faɗin akasin haka: cikin haihuwa 196,608, jeri na 17 ko fiye yana bayyana kusan sau huɗu cikin biyar. Marbe ya gano daidai yadda jerin sakamako na kwatsam ke kama. Epstein (2013) ya sake ba da labarin; shafin Yiwuwar jerin sakamako iri ɗaya a roulette yana ba da nau'in roulette.

Inda kuskuren ba ya aiki
Idan dabaran ya karkata a zahiri, juye-juyen da suka wuce za su ba da bayani, domin za su zama shaida game da ramuka maimakon game da sa'a. Wannan shi ne batun Ɗakin gwajin roulette, kuma yana buƙatar dubban juye-juye daga dabaran ɗaya da aka sani kafin ya faɗi wani abu. Ga jerin ja goma a dabaran da ba a sani ba, bayanin karkata ya fi rashin yiwuwa ƙwarai fiye da bayanin kwatsam.
Tambayoyi
A'a. Yana aiki ga duk wani abu da ake maimaitawa ba tare da dogaro da juna ba: dice, kwabo, lambobin lottery.
Kason za su kusanci juna. Duk da haka sakamakonku zai kasance kusan −2.70% na abin da kuka yi caca da shi.
Babu dokar matsakaita. Akwai dokar manyan lambobi, kuma tana aiki ta hanyar narkewa.
Nishaɗi da lissafi
Ku fara gwada waɗannan da kanku. Kowace mafita a naɗe take; ku buɗe ta idan kun shirya.
Bayan jefa sisi sau 10, ya nuna kai sau 7 da baya sau 3. Bayan jefawa sau 1,000, ya nuna kai sau 517 da baya sau 483. Shin baya ya cim ma kai?
Nuna mafita
Amsa A'a. Tazarar ta ƙaru daga 4 zuwa 34. Ma'aunin rabo ne kawai ya matso kusa da daidai.
Rabon kai zuwa baya ya sauka daga 7/3, kusan 2.33, zuwa 517/483, kusan 1.07. Wannan 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 → ce take aiki.
Tazarar da ke tsakaninsu ta ƙaru, daga 4 zuwa 34. Da sisi na adalci, tazara ta kusan 30 bayan jefawa 1,000 abu ne na yau da kullum: tazarar da aka saba tana ƙaruwa kusan da tushen murabba'i na yawan jefawa, yayin da rabon yake daidaita zuwa 1.
Babu abin da ya janyo baya ya dawo. Ƙarin kai na farko ya narke a cikin jefawa 990 na gaba, ba a gyara shi ba.
Ku jefa sisi sau huɗu kuma ku duba jefawar da ta zo kai tsaye bayan kai kawai. Wane kaso ne daga cikinsu kai ne? Ku lissafa shi ga kowane daga cikin jerika 16 masu yiwuwa kuma ku ɗauki matsakaicin kason. Shin 50% ne?
Nuna mafita
Amsa A'a. 17/42 ne, kusan 40.5%. Idan aka duba ta wannan hanya, sisi na adalci yana kama da yana guje wa kai bayan kai.
Ku jera jerika 16 (H = kai, T = baya). Biyu daga cikinsu, TTTT da TTTH, ba su da kai a cikin jefawa uku na farko, don haka babu abin da yake biyo bayan kai; sai 14 suka rage.
A kowanne, ku ɗauki jefawar da take biyo bayan kai kuma ku nemo kason da yake kai. HHHH yana ba da 3 cikin 3. HHHT yana ba da 2 cikin 3. HTHT yana ba da 0 cikin 2. Idan aka bi duka 14: jerika shida suna ba da 0, huɗu suna ba da 1/2, ɗaya yana ba da 2/3 kuma uku suna ba da 1 (HHHH, THHH da TTHH).
Ku ƙara kason: 4 × 1/2 + 2/3 + 3 = 17/3. Ku raba da 14: 17/42 ≈ 40.5%.
Babu jefawar da take fifita baya; kowace 50–50 ce. Karkatar tana fitowa ne daga ɗaukar matsakaici jeri bayan jeri: jerin da ke cike da kai yana da jefawa masu yawa da za a duba amma ana ƙirga shi sau ɗaya, daidai da jerin da yake da ɗaya kawai. Miller da Sanjurjo (2018) sun nuna cewa binciken hot hand na 1985 ya auna ʼyan wasan ƙwallon kwando ta wannan hanya, don haka ʼyan wasan da suka yi kama da ba su fi dama ta kwatsam yawan jerin nasara ba a zahiri sun ɗan fi.
Ɗan barkwanci

Narkewa, ba gyara ba. Dabaran ba shi da bashin kowa.
Majiyoyi
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Schwartz, D. G. Roll the Bones: The History of Gambling.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Croson, R. & Sundali, J. “The gambler’s fallacy and the hot hand: empirical data from casinos.”
- Chen, D. L., Moskowitz, T. J. & Shue, K. “Decision making under the gambler’s fallacy: evidence from asylum judges, loan officers, and baseball umpires.”
- Clotfelter, C. T. & Cook, P. J. “The ‘gambler’s fallacy’ in lottery play.”
- Gilovich, T., Vallone, R. & Tversky, A. “The hot hand in basketball: on the misperception of random sequences.”
- Miller, J. B. & Sanjurjo, A. “Surprised by the hot hand fallacy? A truth in the law of small numbers.”
- Hacking, I. “The inverse gambler’s fallacy: the argument from design. The anthropic principle applied to Wheeler universes.”