Matsala: dabaran ranar haihuwa
Juye-juye nawa ake yi kafin wata lamba ta maimaita? Ƙasa da yadda yawancin mutane ke zato.
Matsalar ranar haihuwa a dabaran mai ramukakayan aikiRamiƊaya daga cikin ɗakuna 37 masu lamba a kan rotor. Ana kuma kiransa: slot, cup.Karanta gaba ɗaya → 37: bayan juye-juye 8 kacal, damar cewa wata lamba ta riga ta fito sau biyu ta fi rabi. ʼYan wasa suna ɗaukar maimaitawa a matsayin wani tsari; amma su ne ainihin abin da ake tsammani daga dabaran na adalci.
Tambayar
Ku juya dabaran Turai na adalci. Juye-juye nawa ake buƙata kafin yiwuwarlissafiYiwuwaDamar faruwar wani sakamako, daga 0 zuwa 1 ko daga 0% zuwa 100%. Lamba ɗaya: 1/37 = 2.70%.Karanta gaba ɗaya → cewa aƙalla juye-juye biyu sun nuna lamba ɗaya ta wuce rabi? Yawancin ʼyan wasa suna hasashen wani wuri kusa da 18 ko 19, rabin 37. Amsar ita ce 8.
Amsar
Ku lissafa yiwuwar rashin maimaitawa sannan ku cire ta daga ɗaya. Bayan juye-juye k, p(duka sun bambanta) = (37/37)(36/37)(35/37)…((38 − k)/37). Ga k = 8: 1 × 0.973 × 0.946 × 0.919 × 0.892 × 0.865 × 0.838 × 0.811 ≈ 0.44, don haka p(wata maimaitawa) ≈ 0.56. Ga k = 14 ta kai 0.93; ga k = 20 ta kai 0.996. Tunanin zuci yana kuskure ne saboda yawan nau'i-nau'in juye-juye yana girma kamar k², kuma kowane nau'i zai iya yin daidai.
Dalilin da ya sa wannan yake da muhimmanci a teburi
Allon sakamakon lambobi 20 na ƙarshe kusan kullum zai nuna aƙalla maimaitawa ɗaya, sau da yawa biyu ko uku. ʼYan wasa suna ganin “17 ya fito sau biyu a juye-juye goma” sai su ce 17 lamba ce mai “zafi”; lissafin ranar haihuwa yana cewa maimaitawa a cikin juye-juye goma tana faruwa 74% na lokaci, kuma lambar da ta maimaita ta kwatsam ce. Tsarin gaskiya ne; ma'anar ba gaskiya ba ce.
Tambaya ta gaba
Juye-juye nawa ake buƙata kafin wata takamaimar lamba, a ce 17, ta fito sau biyu? Wannan tambaya ce dabam mai amsa mafi girma sosai: jiran da ake tsammani juye-juye 74 ne, kuma yiwuwar 17 biyu a cikin juye-juye 20 kusan 10% ce. Bambancin da ke tsakanin “wata lamba ta maimaita” (juye-juye 8) da “wannan lambar ta maimaita” (juye-juye 74) shi ne gaba ɗaya bambancin da ke tsakanin tsarin da kuke lura da shi bayan abu ya faru da hasashen da kuke yi tun kafin ya faru.
Ku gwada
Ku kunna Wasan gwaji na juye-juye goma sannan ku duba tarihin. Yawanci za a sami maimaitawa. Ku sake kunnawa. Lambar da ta maimaita za ta bambanta. Ku yi haka sau goma sai ku ga matsalar ranar haihuwalissafiMatsalar ranar haihuwaMutane nawa ake buƙata kafin ya fi yiwuwa biyu daga cikinsu su kasance da ranar haihuwa ɗaya, ko juye-juye nawa kafin ya fi yiwuwa wata lamba ta maimaitu. A dabaran mai ramuka 37, juyi 8 ya isa a sami damar da ta fi rabi.Karanta gaba ɗaya →; ku yi shi sau dubu (simulators za su iya) sai ku auna ta.
Ƙa'idar lissafi
Tambayoyi
Daidai, sai dai da ramuka 37 maimakon kwanaki 365. Mutane 23 suna ba da 50% ga ranakun haihuwa; juye-juye 8 suna ba da 56% ga dabaran.
50% bayan kusan juye-juye 19. Har yanzu abu ne na yau da kullum.
A'a. Dubban juye-juye daga dabaran ɗaya za su iya nuna ta; maimaitawa ɗaya ba za ta taɓa nunawa ba.
Nishaɗi da lissafi
Ku fara gwada waɗannan da kanku. Kowace mafita a naɗe take; ku buɗe ta idan kun shirya.
Allon sakamako da ke gefen dabaran na Amurka yana nuna juye-juye 16 na ƙarshe. Yaya yiwuwar babu lambar da ta bayyana sau biyu a kansa?
Nuna mafita
Amsa Kusan 2.5%: kusan allo 39 cikin 40 suna nuna aƙalla maimaitawa ɗaya.
Ku bi juyi bayan juyi. Juyi na farko zai iya zama komai. Na biyu dole ne ya kauce masa: dama 37 cikin 38. Na uku dole ne ya kauce wa duka biyun: 36 cikin 38. Na goma sha shida dole ne ya kauce wa lambobi goma sha biyar: 23 cikin 38.
Ku ninka: 38/38 × 37/38 × 36/38 × … × 23/38 ≈ 0.0246.
Allo marar maimaitawa shi ne wanda ba safai ake gani ba. Allo mai maimaitawa ba ya faɗin komai game da dabaran.
A cikin juye-juye goma na dabaran na Turai, mece ce yiwuwar 17 ya fito sau biyar daidai? Cewa wata lamba, ko wace ce, ta fito sau biyar ko fiye? Kuma cewa wata lamba ta fito aƙalla sau uku a cikin juye-juye huɗu?
Nuna mafita
Amsa 17 sau biyar daidai: 1 cikin 315,576. Wata lamba sau biyar ko fiye: kusan 1 cikin 8,333. Wata lamba sau uku ko fiye a cikin juye-juye huɗu: 1 cikin 349.
17 sau biyar: ku zaɓi waɗanne juye-juye biyar daga cikin goma, hanyoyi 252, kowace da yiwuwar (1/37)⁵ × (36/37)⁵. Gaba ɗaya, 1 cikin 315,576.
Wata lamba: kowace daga cikin lambobi 37 tana da yiwuwar 1 cikin 308,325 ta samun fitowa biyar ko fiye, wanda yake ba da 37 cikin 308,325, ko 1 cikin 8,333. Lamarin da ba safai yake faruwa ba na lambobi biyu kowace da fitowa biyar ana ƙirga shi sau biyu, amma cire shi ba ya canza komai a wannan daidaito.
Sau uku a cikin juye-juye huɗu: ga lamba ɗaya, 4 × 36 + 1 = 145 daga cikin sakamako 37⁴ masu yiwuwa; ga kowace daga cikin lambobi 37, 145/37³ = 1 cikin 349.
Ga dabaran 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, alƙaluman da suka dace su ne 1 cikin 359,275, 1 cikin 9,243 da 1 cikin 368. Tazarar da ke tsakanin “17” da “wata lamba” ita ce matsalar ranar haihuwalissafiMatsalar ranar haihuwaMutane nawa ake buƙata kafin ya fi yiwuwa biyu daga cikinsu su kasance da ranar haihuwa ɗaya, ko juye-juye nawa kafin ya fi yiwuwa wata lamba ta maimaitu. A dabaran mai ramuka 37, juyi 8 ya isa a sami damar da ta fi rabi.Karanta gaba ɗaya → kuma: akwai hanyoyi 37 na yin mamaki.
Ɗan barkwanci
Wani abu yana maimaitawa da wuri. Wannan abin kuwa na kwatsam ne.
Majiyoyi
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Wizard of Odds. “Ask the Wizard”, tambayoyi kan roulette.