Lxeṭṛ n uɣelluy

Q: Anagraw n usnerni yessenqas lxeṭṛ n uɣelluy?

Martingale tessenqas-it i tɣimiyin tiwezlanin, tessali-t aṭas i tɣezfanin. Rrehn ur nettbeddil s uferdis meqqren d abrid n tidet i usenqes-is.

Taprubabilit n usexser n ugerruj-nwen akk uqbel ad tawḍem ɣer yiswi.

Tiririt tawezlant

I rrhun s lexlaṣ imsawi, lxeṭṛ n uɣelluy yeḍfer tasemselt n uɣelluy n umqemmer. S ugerruj n $500, rrhun n $10 d yiswi n $600, ad tawḍem ɣer yiswi qrib 57% n wakud, ad texsrem kullec deg wayen d-yeqqimen.

tasemselt

aɣelluy n umqemmer

p (lexlaṣ imsawi)

18/37

q/p

1,0556

agmuḍ aqbur akk

Pascal–Fermat, 1656

Asteqsi

Tesɛam a n yiferdisenidrimenAferdisRrehn agejdan s wacu i yettxeddem unagraw neɣ amurar. Inagrawen n usnerni ttwaglamen s yiferdisen akken ad ilin i yal ṭṭabla.Ɣer akk → n ugerrujidrimenAgerrujAkk idrimen yettwaǧǧan i wurar. Asefrek n ugerruj d tagrut: acḥal seg-s ara tesrem i yal rrehn d yal taɣimit.Ɣer akk →, tebɣam ad tawḍem ɣer a + b n yiferdisen, s rrehn n yiwen uferdis yal tikkelt ɣef rrehn s lexlaṣ imsawirrehnRrehn s lexlaṣ imsawiAzeggaɣ/aberkan, aryan/uyug, afellay/adday. 18 n wuṭṭunen, ixelleṣ 1:1.Ɣer akk → s taprubabilittusnaktTaprubabilitAzal n yiwen ugmuḍ ad d-yeḍru, seg 0 ɣer 1 neɣ seg 0% ɣer 100%. Yiwen wuṭṭun: 1/37 = 2,70%.Ɣer akk → n rrbeḥ p = 18/37 d taprubabilit n lexsara q = 19/37. D acu i d taprubabilit ad tawḍem ɣer yiswi uqbel ad tawḍem ɣer yilem? Tiririt ufan-tt Pascal, Fermat d Huygens deg yiseggasen n 1650, yerna d yiwen seg yigmaḍ iqburen akk deg taprubabilit.

Blaise Pascal: tamucuha n uḍebsi n rulit

Tasemselt n uɣelluy n umqemmer

P(awaḍ ɣer yiswi) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ). Lxeṭṛ n uɣelluytusnaktLxeṭṛ n uɣelluyTaprubabilit n uxsar n ugerruj meṛṛa send ad tawḍem iswi neɣ amḍan n tuzzyiwin.Ɣer akk → d yiwen ma nekkes-as aya. S q/p = 19/18 ≈ 1,0556, assaɣ yettnerni s wudem asemyif d umḍan n yiferdisen, ihi tasemselt tettɛaqab iferdisen imecṭaḥ s ṣṣeḥḥa: mi ara ilaq ugar n rrhun akken ad tawḍem ɣer yiswi, ugar n tikkal ara txelṣem tabzert n 2,70% deg ubrid.

agerrujaferdisiswirrhun alamma d iswip(awaḍ ɣer yiswi)lxeṭṛ n uɣelluy

$500$10$60010≈ 57%≈ 43%

$500$50$6002≈ 79%≈ 21%

$500$10$1 00050≈ 6%≈ 94%

$500$100$1 0005≈ 43%≈ 57%

$500$500$1 000148,6%51,4%

$100$10$20010≈ 37%≈ 63%

Rrhun s lexlaṣ imsawi, rrehn ur nettbeddil, war la partage. Tamsiḍent n lxeṭṛ n uɣelluy txeddem leḥsab i yal amḍan.

Taɣuri-ines

Iferdisen imecṭaḥ ttḍurrun. Aneɛruḍ n usnerni s sin n $500 s rrhun n $10 yettrennu 6% n wakud; s yiwen n rrehn n $500 yettrennu 48,6% n wakud. Urar n tebɣest, drus n rrhun imeqqranen, yessimɣur taprubabilit n wawaḍ ɣer yiswi deg wurar s wazal yettwaṛǧan ddaw n yilem, acku yessenqas amḍan n tikkal i tettwasnas lmaziyya. Yessimɣur daɣen tavaryanstusnaktTavaryansAktazal n wamek i d-ttwazzaren igmaḍ i tama n wazal yettwaṛǧan. Meqqret i rrehn ɣef yiwen wuṭṭun, meẓẓiyet i rrhun s lexlaṣ imsawi.Ɣer akk → n wayen ara awen-yeḍrun, yerna yessekfay tameddit s tɣawla. Alugen-a n tusnakt semman-as «urar n tebɣest d win yifen» yerna sbeggnen-t Dubins d Savage deg 1965.

Taprubabilit n uselkem n $500 ɣer $1 000 ɣef uzeggaɣ%

yiwen n rrehn n $500

48,6%

rrhun n $100

≈ 43%

rrhun n $50

≈ 37%

rrhun n $20

≈ 21%

rrhun n $10

≈ 6%

rrhun n $5

≈ 0,5%

024,3048,6%

Mi ara ilaq ugar n rrhun, ugar n tikkal i tettwaxelleṣ lmaziyya. Urar n tugdi yettkellif aṭas.

Anida i d-yebda wugur

Asteqsi-a yezwar rulit. Deg 1656 Pascal yefka i Fermat ugur: sin n yimuraren ttḍeggiren kraḍ n zzhaṛ, yal yiwen yebda s 12 n yijiṭunen n usiḍen; aḍeggeṛ n 11 yessenkaz ajiṭun ɣer yiwet n tama, aḍeggeṛ n 14 ɣer tayeḍ. Sin yid-sen fran-t, Huygens, i yesslen yis-s seg tebṛat n umusnaw n tusnakt Pierre de Carcavi, yessiggez-it deg 1657 am wugur wis semmus n udlis amezwaru ɣef taprubabilit, s tririt war abeggen. Ḥdac yezmer ad yettwaḍeggeṛ s 27 n yiberdan, arbaṭac s 15 kan, ihi yal ajiṭun yettenkaz ɣer tama n umurar amezwaru 9 n tikkal seg 14, yerna tasemselt d-yezwaren, s 12 n yijiṭunen i yal yiwen, terra-t qrib 1 157 n tikkal s taprubabilit meqqren ad yerbeḥ kullec ugar n usexser n kullec. Lmaziyya n 2,70% d tamecṭuḥt aṭas, maca yiwet n tsemselt tesskan ayen txeddem deg waṭas n rrhun.

Deg wugur n Pascal, yal aḍeggeṛ n kraḍ n zzhaṛ yessenkaz yiwen ujiṭun gar yimuraren.

Rrehn n tebɣest n tidet

Deg yebrir 2004, Ashley Revell, d ilemẓi Aglizi, yezzenz qrib kullec ayen yesɛa, ula d llebsa-s, yewwi $135 300 ɣer Plaza deg Las Vegas. Deffir kra n rrhun imecṭaḥ, yessers azal akk ɣef uzeggaɣrrehnAzeggaɣRrehn s lexlaṣ imsawi ɣef 18 n wuṭṭunen izeggaɣen. S tefransist: rouge.Ɣer akk →, yiwet n tikkelt. TakurtibenkanTakurtTakurt tamecṭuḥt i yettwaḍeqqren deg ubrid mgal aqerru yettezzin. Tikurin n tura d ivorine neɣ d Teflon, 18–21 mm.Ɣer akk → teɣli ɣef 7 azeggaɣ, yeffeɣ s $270 600. Ɣef uḍebsi s sin yilmawenaḍebsiIlemTaxxamt tazegzawt tasuft ur yeṭṭif ula d yiwen n rrehn s lexlaṣ imsawi, n ddzina neɣ n tgejdit; d aɣbalu n lmaziyya n lkazinu. Daɣen: yiwen yilem, 0.Ɣer akk →, tuzzya-nni d taprubabilit n 47,37%, tkellef-as $7 121 s ulemmas, lmaziyya n 5,26% ɣef yiwen n rrehn. Lukan yebḍa-t ɣef rrhun n $10, iswi-nni ad yili qrib ur yettwaweḍ ara. Urar n tebɣest ur yettarra ara rrehn yelha; yettarra-t d awezlan akken yezmer.

Agerruj akk n Ashley Revell yella ɣef yiwet n tuzzya n uḍebsi s sin yilmawen.

Lxeṭṛ n uɣelluy war iswi

Ma yella turarem kan alamma texsrem kullec neɣ teḥbsem deffir N n tuzzyiwin, asteqsi yuɣal d «d acu i d taprubabilit ad yaweḍ ugerruj-iw ɣer yilem deg N n tuzzyiwin?». Ulac tasemselt tamedlant i yal N, maca isimulaturen n usmel-a ḥesben-tt: agerruj n $500 s $10 ɣef uzeggaɣ qrib werǧin yettɣelli deg 100 n tuzzyiwin (ddaw 1%), yettɣelli qrib 9% n wakud deg 500 n tuzzyiwin, qrib 95% n wakud deg 5 000. War alugen n usbeddi taprubabilit tettqarrab ɣer 100%, d wagi i d lqem n uɣawas aɣezfan n lmaziyya n lkazinutusnaktLmaziyya n lkazinuAḥric n yal rrehn i teṭṭef lkazinu s ulemmas: 2,70% deg rulit tawrupit.Ɣer akk →.

Mraw n tɣimiyin, agerruj n $500, $25 ɣef uzeggaɣ, 200 n tuzzyiwinagmuḍ mgal tazwara

1 000 n tɣimiyin × 500 n tuzzyiwin · amidadi p10/p25/p75/p90, anili afsas azal n tlemmast, 81 n yimedyaten ifsasen · sekcmet taɣerdayt deg udlif i yiswiren n yal tuzzya

p10 −$500
p25 −$500
azal n tlemmast −$500
p75 +$0
p90 +$350

1 000 n tɣimiyin × 500 n tuzzyiwin · agellus y yettwagzem ɣer yiswiren 1–99 (seg uzal amecṭuḥ akk −$500 ar umeqqran akk +$1 500) · 500 n tuzzyiwin · tazwara $25 · lexlaṣ imsawi

Ijerriḍen i yewwḍen ɣer wadda ɣlin. S uferdis amecṭuḥ ad ṭṭulen ugar, ad fakken deg yiwen umkan.

Amlili deg uɣawas aɣezfan: alugen n wuṭṭunen imeqqranen deg rulit

Acḥal ara yeṭṭul, d wacḥal ara yezwir

Sin n yigmaḍ iqburen nniḍen rran-d ɣef yisteqsiyen i ttesteqsayen yimuraren s tidet. War iswi, rrhun ur nettbeddil s lexlaṣ imsawi ṭṭulen s ulemmas 37 n tuzzyiwin i yal aferdis n ugerruj ɣef uḍebsi awrupi d 19 ɣef umarikani, acku yal tuzzya tettawi 1/37 n uferdis s ulemmas (1/19 ɣef uḍebsi amarikani): 100 n yiferdisen ṭṭulen 3 700 n tuzzyiwin s ulemmas, ɣas akken azgen n tɣimiyin am tigi fukkent deg 3 134 (walit Teɣzi n tɣimit d ssuma-ines). Yerna akken yebɣu yili ugerruj meqqer, taprubabilit ɣef uḍebsi awrupi ad yili umurar yiwet n tikkelt s k n yiferdisen n rrbeḥ d (18/19)ᵏ. D 95% i yiwen uferdis d 58% i mraw, maca ulac agerruj, akken yebɣu yili d aqeɛɛar, i d-yettaken taprubabilit n wazgen ad yili umurar yiwet n tikkelt s 13 n yiferdisen n rrbeḥ; ɣef uḍebsi amarikani taprubabilit d (9/10)ᵏ, tettṣubbu ddaw wazgen deg 7 n yiferdisen.

iferdisen n rrbeḥ, deg yal tanadditTawrupitTamarikanit

194,7%90,0%

576,3%59,0%

1058,2%34,9%

1349,5%25,4%

2033,9%12,2%

506,7%0,5%

1000,45%0,003%

Rrhun s lexlaṣ imsawi, rrehn ur nettbeddil, agerruj meqqren akken werǧin ad yekfu. Taprubabilit d (18/19)ᵏ ɣef uḍebsi s yiwen yilem d (9/10)ᵏ ɣef win s sin yilmawen.

Teɣzi n tɣimit d ssuma-ines Rulit tamarikanit: aḍebsi d yilugan

Agerruj i taprubabilit n wazgen

Qelbet asteqsi: acḥal i ilaq ad d-tawim akken ad tesɛum ma drus taprubabilit n wazgen ad terbḥem azal yettwasbedden uqbel ad texsrem kullec? I rrbeḥ amecṭuḥ, mačči aṭas. I win meqqren ugar, agerruj ilaqen yettali s tɣawla, yerna seg yiwet n tnaddit ulac agerruj i yekfan, acku ula d win ur nfukk ara yettak ddaw n taprubabilit n wazgen ad yili yiwen yiwet n tikkelt s rrbeḥ-nni. Taylor (2021) yenna-t s wudem nniḍen: ɣef uḍebsi amarikani, akken ad tesɛum taprubabilit n wazgen ad tawḍem ɣer 100 n yiferdisen, ilaq ad tebdum s 94, ad tɣamrem s 94 akken ad terbḥem 6. Ɣef uḍebsi awrupi ad tesrem 88.

akken ad terbḥemTawrupitTamarikanit

1 n uferdis22

2 n yiferdisen33

3 n yiferdisen45

4 n yiferdisen68

5 n yiferdisen712

6 n yiferdisen921

7 n yiferdisen12ulac agerruj i yekfan

8 n yiferdisen15ulac agerruj i yekfan

10 n yiferdisen24ulac agerruj i yekfan

12 n yiferdisen46ulac agerruj i yekfan

13 n yiferdisen neɣ ugarulac agerruj i yekfanulac agerruj i yekfan

Agerruj amecṭuḥ akk, s yiferdisen, i d-yettaken ma drus taprubabilit n wazgen ad yettwarbeḥ wazal uqbel ad yexser ugerruj, s rrhun ur nettbeddil s lexlaṣ imsawi. Leḥsab-nneɣ seg tsemselt d-yezwaren.

Tasirt n leqdic

Fernet ayen i tebɣam ad txedmem. Ma yella iswi d rrbeḥ yettwasbedden, rhent aṭas, drus n tikkal. Ma yella iswi d tameddit taɣezfant, rhent drus, qeblet belli lmaziyya ad tawi 2,70%-ines seg umḍan n rrhun meṛṛaidrimenAmḍan n rrhun meṛṛaTamerna n yidrimen yettwarhen deg yiwet n tɣimit neɣ deg yiwen wakud, s rrbeḥ i yettuɣalen yettwarhen. Daɣen: handle, action.Ɣer akk → ameqqran. Ma yella iswi d sin, ilaq yiwen ad yefk abrid, s umata d iswi.

Tasemselt

P(iswi) = (1 − (q/p)ᵃ) / (1 − (q/p)ᵃ⁺ᵇ)

lxeṭṛ n uɣelluy = 1 − P(iswi)

q/p = 19/18 i lexlaṣ imsawi

alemmas n tuzzyiwin alamma d aɣelluy, war iswi = 37 × iferdisen

P(yiwet n tikkelt s k n yiferdisen n rrbeḥ) = (18/19)ᵏ

Isteqsiyen

Anagraw n usnerni yessenqas lxeṭṛ n uɣelluy?

MartingaletastratijitMartingaleAnagraw n usnerni i yessneggmen rrehn deffir yal lexsara. Walit taggayt n tstratijit i wayen i yettxemmem d wayen ur yettxemmim ara.Ɣer akk → tessenqas-it i tɣimiyin tiwezlanin, tessali-t aṭas i tɣezfanin. Rrehn ur nettbeddiltastratijitRrehn ur nettbeddilRrehn n yiwen wazal deg yal tuzzya. D aswir ɣer wacu i yettwaserwas yal anagraw n usnerni.Ɣer akk → s uferdis meqqren d abrid n tidet i usenqes-is.

Ma d rrhun ɣef yiwen wuṭṭun?

Tasemselt d-yezwaren d tin n lexlaṣ imsawi. I rrhun s tavaryans meqqren, aɣelluy yesɛa taprubabilit meqqren s rrhun imecṭaḥ, iswi yettwaweḍ s wudem yettneggizen; sseqdcet tamsiḍent.

Ayɣer q/p d amḍan agejdan?

Yal tikli ɣer yiswi tesɛa taprubabilit drus cwiṭ n yal tikli ɣer uɣelluy; assaɣ yettwasgat i yal tikli.

Urar s tusnakt

Ɛerḍet-tent s yiman-nwen qbel. Yal tifrat tettwaffer; ldit-tt mi ara tilim twejdem.

Tesɛam $100 u tebɣam $1 000, s rrehn ɣef uzeggaɣrrehnAzeggaɣRrehn s lexlaṣ imsawi ɣef 18 n wuṭṭunen izeggaɣen. S tefransist: rouge.Ɣer akk → kan. Aɣawas amezwaru: rhent $1 yal tikkelt. Aɣawas wis sin: rhent ayen meṛṛa i tesɛam ar ad tɛeddim $500, syin ayen yettusran kan akken ad tawḍem $1 000. Anwa aɣawas yelhan ugar, s wacḥal?
Sken-d tifrat
Tiririt Aɣawas wis sin, s waṭas. Yettawweḍ ɣer $1 000 qrib 9,0% n wakud, qrib 10%-nni n lḥeqq. Aɣawas amezwaru yeddu qrib 7 n tikkal deg 10²².
Deg wurar n lḥeqqtusnaktUrar n lḥeqqUrar anda ulac win ɣur-s lmaziyya: yal rrehn azal-is yettwaṛǧan d ilem.Ɣer akk →, ad tawḍem $1 000 seg $100 yiwet n tikkelt deg mraw, akken yebɣu yili wayen txeddmem, acku azal alemmas n yedrimen-nwen yeqqim ɣef $100.
Lmaziyya teṭṭef 2,70% n yal rrehn, ihi ma drus wayen trehnem s umata, ad teqqimem qrib yiwet deg mraw. Aɣawas wis sin yesbadu ayen deg kra n rrhun: $100 yuɣal d $200, $400, $800, syin $200 qudden. Yeddu qrib 9,0% n wakud. Aɣawas amezwaru yesra igiman n rrhun qbel ad yekfu wbrid, u lmaziyya, tamecṭuḥt ɣef yal yiwen, temrennu ar uɣelluy qrib yettwaḍemnen.
D agmuḍ n Dubins–Savage s talɣa-s tafessast akk (Epstein, 2013): deg wurar i yesmenyafen lkazinu, rhent ayen akk i tesrem, drus n tikkal akken tzemrem.
Tesɛam $500 u tebɣam $1 000, ad tḥebsem mi ara ten-tawim. Tefrenem yiwet n tewsit n rrehn u tettɛawadem-tt ar ad terbeḥ yiwet n tikkelt neɣ ad fakken yedrimen. Anta tawsit i yettaken tagnit yelhan akk?
Sken-d tifrat
Tiririt Yiwen n rrehn n $500 ɣef uzeggaɣ: 48,6%. Yal aɣawas i yebḍan idrimen ɣef rrhun imecṭaḥ d win n diri ugar.
Fernet yal adrim n rrehn akken yiwen n rrbeḥ ad yeqqed: $30 ɣef yiwen wuṭṭunrrehnYiwen wuṭṭunYiwen wuṭṭun kan, ixelleṣ 35:1.Ɣer akk → yerbeḥ $1 050, $60 ɣef sin wuṭṭunenrrehnSin wuṭṭunenSin wuṭṭunen yemyuḍanen, ixelleṣ 17:1.Ɣer akk → yerbeḥ $1 020, akka akka. Syin ḥesbet taprubabilit ad yerbeḥ ma drus yiwen n rrehn qbel ad fakken yedrimen. Sṭac n rrhun n $30 ɣef yiwen wuṭṭun xesren akk s tprubabilit n (36/37)¹⁶ = 64,5%, ihi ttawḍen ɣer $1 000 35,5% n wakud.
Tafelwit tesskan-d yal rrehn. Azeggaɣ d netta i yelhan akk acku yettawweḍ s yiwen n rrehn, ihi lmaziyya tettwaxelleṣ yiwet n tikkelt, ɣef $500. Sin yizirigenrrehnSin yizirigenSḍis n wuṭṭunen deg sin yijerriḍen yemyuḍanen, ixelleṣ 5:1. S tefransist: sixain.Ɣer akk → d wis sin acku yesra kraḍ kan. Yiwen n rrehn ɣef ddzinarrehnDdzina1–12, 13–24 neɣ 25–36, ixelleṣ 2:1.Ɣer akk → d win n diri akk, s tprubabilit n 12/37; ula ma yettwaurar s yiferdisenidrimenAferdisRrehn agejdan s wacu i yettxeddem unagraw neɣ amurar. Inagrawen n usnerni ttwaglamen s yiferdisen akken ad ilin i yal ṭṭabla.Ɣer akk → n $250 ar ad tawim $1 000 neɣ ugar, ddzina tettawweḍ 46,4% n wakud kan.
Bollman (2024) yebna tafelwit-nni yiwet i uḍebsi amarikani. Ajerriḍ-is n ddzina yesɛa tuccḍa: ilaq ad yili d 12/38 = 31,6%.
rrehnadrim n rrehn × amḍan n rrhuntaprubabilit n uwaḍ ɣer $1 000
Yiwen wuṭṭun$30 × 1635,5%
Sin wuṭṭunen$60 × 835,9%
Izirig$90 × 534,5%
Tiɣmert$125 × 436,7%
Sin yizirigen$166, $166, $16841,2%
Ddzina$500 × 132,4%
Azeggaɣ$500 × 148,6%
Trehnem yiwen n uferdis ɣef uzeggaɣ, tuzzya deffir tuzzya, s ugerrujidrimenAgerrujAkk idrimen yettwaǧǧan i wurar. Asefrek n ugerruj d tagrut: acḥal seg-s ara tesrem i yal rrehn d yal taɣimit.Ɣer akk → ur nettfakka ara. Deffir tuzzya tamezwarut, tesɛam yiwen n uferdis n rrbeḥ neɣ yiwen n lexsara. Acḥal d taprubabilit ad tuɣalem yiwet n tikkelt ɣer umsawi s tidet?
Sken-d tifrat
Tiririt 36 deg 37, 97,3%. Ɣef uḍebsi amarikani d 36 deg 38, 94,7%.
Ma terbeḥ tuzzya tamezwarut (18 n tikkal deg 37), tesɛam yiwen n uferdis n rrbeḥ, u lmaziyya tjebbed yal taɣimit war taggara ɣer wadda, ihi ɣas ma ɛeṭṭlen, ad tɛeddim ɣef umsawi: d tidet yettwaḍemnen. Ma texser (19 n tikkal deg 37), ilaq ad d-terrem yiwen n uferdis, u taprubabilit ad tilim yiwet n tikkelt yiwen n uferdis zdat d 18/19 (walit tafelwit nnig-a).
Akk: 18/37 + 19/37 × 18/19 = 18/37 + 18/37 = 36/37.
S umata, taprubabilit d 1 ma nekkes amgired gar tprubabilit n rrbeḥ d tin n lexsara n tuzzya: 1 − 1/37 ɣef uḍebsi awrupi, 1 − 2/38 ɣef win amarikani. Deg wurar war ilemaḍebsiIlemTaxxamt tazegzawt tasuft ur yeṭṭif ula d yiwen n rrehn s lexlaṣ imsawi, n ddzina neɣ n tgejdit; d aɣbalu n lmaziyya n lkazinu. Daɣen: yiwen yilem, 0.Ɣer akk → amgired d ilem u taprubabilit d 1: tanqiḍt tettuɣal yal tikkelt ɣer umsawi. Epstein (2013) yefka-d alugen amatu.
Tamsalt n Pascal n 1656: sin yimuraren ḍeqqren kraḍ n zzhaṛ, yal yiwen yebda s 12 n yijiṭunen n leḥsab. Yal 11 tessmutti ajiṭun ɣer umurar amezwaru, yal 14 tessmutti yiwen ɣer wis sin; tigrin nniḍen ur ttwaḥsabent ara. Acḥal d taprubabilit ad yerbeḥ umurar amezwaru 24-nni meṛṛa? U ma yella trehnem deg wadeg-is 1 n uferdis ɣef uzeggaɣ yal tuzzya mgal amdakkel yuɣalen d lbanka, 12 n yiferdisen yal yiwen, acḥal d taprubabilit ad terbḥem 24-nni meṛṛa?
Sken-d tifrat
Tiririt Amurar amezwaru yerbeḥ qrib 1 157 n tikkal deg 1 158. Kunwi, ɣef uzeggaɣ, ad terbḥem 24-nni n yiferdisen qrib 34% n wakud kan.
Kraḍ n zzhaṛ ttak-d 11 s 27 n tarrayin d 14 s 15 n tarrayin, ihi yal ajiṭun yettwasmuttin yettruḥ ɣer umurar amezwaru 27 n tikkal deg 42, 9 n tikkal deg 14. Imi yal tama tebda s 12, taprubabilit n tuṭṭfa n kullec d 1/(1 + r¹²), anda r d assaɣ n tprubabilit n tama nniḍen ɣef tin-nwen.
I umurar amezwaru r = 5/9, u (5/9)¹² ≈ 0,00086, ihi taprubabilit d qrib 0,99914: assaɣ n 9¹² ɣef 5¹², neɣ 282 429 536 481 ɣef 244 140 625, qrib 1 157 ɣef 1. Huygens yessuffeɣ-d imḍanen-a s tidet deg 1657.
Ɣef uzeggaɣ, r = 19/18 s ssebba n yilem, u (19/18)¹² ≈ 1,913: taprubabilit-nwen d 1/2,913 ≈ 34,3%. Aṣurdi yili yettak-d 50%. Lmaziyya n 2,70% ɣef yal rrehn temrennu, deg waṭas n rrhun i yettusran, ar umgired n 16 n tenqiḍin.

Taḍsa i usgunfu

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Iɣbula

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1. Wiley, taẓrigt tis 3, 1968. Aɣelluy n umqemmer, ixef XIV.
Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed. Academic Press, 2013.
Taylor, D. G. Games, Gambling, and Probability: An Introduction to Mathematics, 2nd ed. CRC Press, 2021.
Carcavi, P. de. Tabṛat i Huygens, 28 ctembeṛ 1656, ɣef wugur i yefka Pascal i Fermat. Œuvres complètes de Christiaan Huygens, alk. 1, 1888, sb. 492–494.
Huygens, C. De Ratiociniis in Ludo Aleae. 1657. Tasleḍt tamezwarut yettwasiggzen n wazal yettwaṛǧan.
Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer, 2010. Ixef 13, rulit.
Reuters. “Briton bets it all on roulette spin, and wins.” Yebrir 2004.
Las Vegas Sun. “Man wagers life’s worth.” 12 yebrir 2004.

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