Mavahe Fakasitanate ʻi he Lulēti
Ko e mavahe fakasitanate ʻi he lulēti, ko e aka sikuea ʻo e vēlianimatematikaVēlianiKo ha fua ʻo hono mafola ʻo e ngaahi ola takatakai ʻi he mahuʻinga ʻamanekina. Māʻolunga ki he peti fika ʻe taha, māʻulalo ki he ngaahi peti totongi tatau.Lau kotoa →, ko e mamaʻo angamaheni ia ʻi ha ola mo hono ʻamanekina.
Ki he $1 naʻe peti, ko e σ ko e 5.84 ki ha peti fika ʻe taha pea 1.00 ki he kulokula. Fakaliunga ʻaki ʻa e peti mo e aka sikuea ʻo e lahi ʻo e ngaahi vilo ki ha taimi vaʻinga.
Lao faingofua
Ko e meimei vahe ua ʻi he tolu ʻo e ngaahi taimi vaʻinga ʻoku tō ʻi he σ ʻe taha mei he ʻamanekina pea ko e 95% ʻi he ua. Ki ha vilo kulokulapetiKulokulaPeti totongi tatau ki he fika kulokula ʻe 18. Faka-Falanisē: rouge.Lau kotoa → ʻe 100 ʻi he $10 (ʻamanekina −$27, σ = $100) ko e lahi taha ʻo e ngaahi taimi vaʻinga ʻoku tō ʻi he vahaʻa ʻo e −$127 mo e +$73, pea meimei kotoa ʻi he vahaʻa ʻo e −$227 mo e +$173. Ko ha ola ʻi tuʻa ʻi he vahaʻa ko ia ʻoku ʻikai ko ha fakamoʻoni ia ki ha meʻa tukukehe pē ʻoku fai ia ʻe ha taimi vaʻinga ʻe taha ʻi he uofulu.
Ko e hā ʻa e √N
ʻOku tupu ʻa e mole ʻamanekinamatematikaMole ʻamanekinaKo e mahuʻinga ʻamanekina liunga ʻaki ʻa e fakakātoa ʻo e paʻanga naʻe peti ʻaki; ko e ʻavalisi ʻo e paʻanga ʻoku mole ʻi ha taimi vaʻinga pe ha vahaʻa taimi.Lau kotoa → ʻo fakatatau ki he N; ʻoku tupu ʻa e mafola ʻo fakatatau ki he √N. Ki ha N siʻi ʻoku pule ʻa e mafola pea ko e ngaahi taimi vaʻinga ʻoku hangē ha lī paʻanga ʻoku kiʻi hehema. Ki ha N lahi ʻoku lele vave ange ʻa e mole ʻi he mafola. Ko e feituʻu ʻoku na fetaulaki ai, ʻa ia ʻoku tatau ai ʻa e mole ʻamanekina mo ha mavahe fakasitanatematematikaMavahe fakasitanateAka ua ʻo e vēliani; ko e mamaʻo angamaheni ʻo ha ola mei he mahuʻinga ʻamanekina.Lau kotoa → ʻe taha, ʻoku ʻi he N = (σ/tuʻunga lelei)² = (1/0.027)² ≈ vilo ʻe 1,370 ki he peti totongi tatau pea (5.84/0.027)² ≈ 46,800 ki he ngaahi peti fika ʻe tahapetiPeti fika ʻe tahaFika pē ʻe taha, totongi ʻo e 35:1.Lau kotoa →. ʻI lalo ʻi he ngaahi fika ko ia, ʻoku fakapapau ʻe he monūʻia; ʻi ʻolunga ai, ʻoku fakapapau ʻe he tuʻunga lelei.
Ko hono ngāueʻaki
Ke fakamaauʻi pe naʻe kehe ha taimi vaʻinga, fakafuofua ʻa e (ola − EV)/σ. Ko ha tokotaha peti kulokula ʻi he $10 ʻoku ne fakaʻosi ʻa e vilo ʻe 100 ʻi he +$150 ʻoku ʻi he (150 + 27)/100 = 1.77σ, ko ha ola ʻe taha ʻi he 25: monūʻia, ka ʻoku ʻikai ko ha meʻa makehe. Ko ha tokotaha peti fika ʻe taha ʻi he +$150 tatau ʻoku ʻi he 0.3σ: ko ha pō angamaheni. Ko e ola tatau ʻi he tola ʻoku kehekehe hono ʻuhinga ʻi he ngaahi peti kehekehe.
Fehalaaki fakasitanate ʻo ha vahe
ʻOku fakahā ʻe he fika tatau pe ko e vilo ʻe fiha ʻoku fiemaʻu ke sio ai ki he tuʻunga lelei ʻi he ngaahi fakamatala. Ko e fehalaaki fakasitanate ʻo ha vahe ikuna kuo sio ki ai ko e √(p(1 − p)/N). Ki he kulokula mo e N = 1,000: √(0.4865 × 0.5135 / 1000) ≈ 1.6%. Ko e vā ʻi he vahaʻa ʻo ha lī paʻanga faitotonu mo e kulokula ko e 1.35%, ko ia hili ha vilo ʻe 1,000 ʻoku teʻeki lava ke ke fakafaikehekeheʻi kinaua ʻi he tui papau. Hili ha 10,000 ko e fehalaaki fakasitanate ko e 0.5% pea ʻoku ke lava. Ko e ʻuhinga foki ia ʻoku fiemaʻu ai ha vilo lahi ʻaupito ke ʻiloʻi ha hehema ʻo e veʻeteka.
Fakatokanga ki he hehema ʻo e fōtunga
Ko e ngaahi ola ʻo e peti fika ʻe taha ʻoku mamaʻo mei ha tufaki angamaheni (normal) ʻi ha taimi vaʻinga nounou: ko e lahi taha ʻo e ngaahi taimi vaʻinga ko ha −$10 maʻu pē ʻi he vilo mo ha ngaahi hopo +$350 ʻi he taimi ʻe niʻihi. ʻOku fakamatalaʻi totonu ʻe he lao ±1σ ʻa e mafola ka ʻoku ʻikai lelei ʻa e fōtunga kae ʻoua kuo aʻu ʻa e N ki he ngaahi teau. Ki he ngaahi peti totongi tataupetiPeti totongi tatauKulokula/ʻuliʻuli, taʻehoa/hoa, māʻolunga/māʻulalo. Fika ʻe 18, totongi ʻo e 1:1.Lau kotoa → ʻoku lelei ʻa e fakafuofua angamaheni mei he meimei vilo ʻe 30.
Fomula
Ngaahi fehuʻi
Ko ha fakamoʻoni ia naʻa ke vaʻinga ʻi ha taimi vaʻinga ʻe meimei 40, pe naʻa ke ʻi he taha monūʻia ʻi he fāngofulu. Toe fai ia.
ʻOku siʻisiʻi ange ʻene kalakala: ko e fehalaaki fakasitanate ʻo e ʻavalisi ko e σ/√N. ʻI he N = 10,000 ko e 1% ia ʻo e peti.
Vaʻinga mo e Matematika
ʻUluaki feinga ke ke fakaleleiʻi ʻe koe. Kuo pelu ʻa e tali takitaha; fakaava ia ʻi hoʻo mateuteu.
ʻI he vilo ʻe 100 ʻo ha veʻeteka ʻIulope, ʻoku ʻi muʻa ha iunitipaʻangaIunitiKo e peti tefito ʻoku ngāueʻaki ʻe ha sisitemi pe ha tokotaha vaʻinga. ʻOku fakamatalaʻi ʻa e ngaahi fakalakalaka peti ʻaki ʻa e iuniti koeʻuhi ke ngāue ʻi ha tēpile pē.Lau kotoa → ʻe taha ki he kulokulapetiKulokulaPeti totongi tatau ki he fika kulokula ʻe 18. Faka-Falanisē: rouge.Lau kotoa → ʻi ha ikuna ʻe 51 pe lahi ange, pea ʻoku ʻi muʻa ha iuniti ʻe taha ki ha peti fika ʻe tahapetiPeti fika ʻe tahaFika pē ʻe taha, totongi ʻo e 35:1.Lau kotoa → kapau ʻe hā hono fika tuʻo 3 pe lahi ange. Ngāueʻaki ʻa e kalava fakafaʻafaʻa (bell curve) ke fakafuofua ʻa e ongo ngalingali, pea fakatatau leva kinaua mo e ngaahi tali tonu.
Fakahā ʻa e tali
Tali Kulokula: ʻoku ʻomi ʻe he kalava fakafaʻafaʻa ʻa e 35.6% pea ko e tali tonu ko e 35.5%. Ko e peti fika ʻe taha: ʻoku ʻomi ʻe he kalava fakafaʻafaʻa ʻa e 55.0% pea ko e tali tonu ko e 50.9%. ʻOku ngāue ʻa e founga nounou ki he kulokula pea hala ki he fika ʻe taha.
Kulokula: ko e ʻavalisi ʻo e ngaahi ikuna ko e 100 × 18/37 = 48.6, mo ha mafolalahi ʻo e √(100 × 18/37 × 19/37) = 5.0. Ko e ʻi muʻa ʻoku ʻuhinga ia ki he 51 pe lahi ange; ko e māka vaeua, 50.5, ʻoku mafolalahi ʻe 0.37 ia ʻi ʻolunga ʻi he ʻavalisi, pea ʻoku tuku ʻe he kalava fakafaʻafaʻa ʻa e 35.6% ʻi ʻolunga ai. Ko hono tānaki ʻo e ngaahi ngalingali tonu ʻo e ikuna ʻe 51, 52 … 100 ʻoku ʻomi ai ʻa e 35.5%.
Peti fika ʻe taha: ko e ʻavalisi ʻo e ngaahi tonu ko e 100/37 = 2.70, mo ha mafolalahi ʻo e √(100 × 1/37 × 36/37) = 1.62. Ko e tonu ʻe tolu pe lahi ange ʻoku ʻuhinga ia ki he ʻolunga ʻi he 2.5, mafolalahi ʻe 0.12 ʻi lalo ʻi he ʻavalisi: 55.0% ʻaki ʻa e kalava fakafaʻafaʻa. ʻI he tonu, ko e 1 toʻo ʻa e ngaahi ngalingali ʻo e tonu ʻe 0, 1 mo e 2, 6.5% + 17.9% + 24.7%, ʻa ia ʻoku toe ai ʻa e 50.9%.
ʻOku fiemaʻu ʻe he kalava fakafaʻafaʻa ha ngaahi ola lahi ʻi he tafaʻaki fakatouʻosi ʻo e ʻavalisi. ʻOku ʻi ai ia ʻi he kulokula hili ʻa e vilo ʻe 100; ka ko ha fika ʻoku hā siʻi hifo ʻi he tuʻo tolu ʻi he ʻavalisi ʻoku ʻikai. ʻOku toe ʻomi foki ʻe he kalava ʻa e 8.7% ki he “ʻikai ha tonu”, ka ko e ngalingali moʻoni ko e 6.5%. ʻOku tuku ʻe Taylor (2021) ʻa e binomial mo e kalava fakafaʻafaʻa ʻo fehoanaki ʻi he founga tatau.
Meʻa fakaoli
Ko e taimi vaʻinga ʻe taha ʻi he uofulu ʻoku ʻi tuʻa ʻi he sikima ʻe ua. ʻOku ʻi ai maʻu pē ha taha ʻi he iku.
Ngaahi Maʻuʻanga
- 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.