Chi-Square Test

Ise nikaalna

Har pocketsaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → ka expected count n/37 hai. Har pocket ke liye observed aur expected ke fark ka square nikaaliye aur expected se bhaag dijiye; 37 terms jodiye. Fair wheel average mein lagbhag 36 (degrees of freedomlabDegrees of freedomKisi statistic mein independent values ki sankhya. 37 pockets par chi-square test ke liye yeh 36 hai, kyunki counts ka jod total ke barabar hona chahiye.Poora padhein →) ki values deta hai, lagbhag 8.5 ke standard deviationganitStandard deviationVariance ka square root; expected value se kisi result ki aam doori.Poora padhein → ke saath. 51.0 se upar 5% level par significant hai; 58.6 se upar 1% par. Wheel Bias Analyzer statistic aur regularized incomplete gamma function se exact p-valuelabp-valueAgar wheel fair hota, to observed deviation jitna ya usse bada deviation dekhne ki probability.Poora padhein → nikaalta hai.

Ek hal kiya hua udaharan

1,000 spins. Chhattis pockets 20 se 34 hits ke beech aate hain; pocket 8 ke 41 hain. Expected 27.03 hai. Akela pocket-8 term (41 − 27.03)²/27.03 = 7.2 hai. Maan lijiye baaki 36 terms ka jod 38 hai (aam baat). 36 df par χ² = 45.2, p ≈ 0.14. Significant nahi: fair wheel aisa saat mein ek baar karta hai. Pocket 8 ek ummeedwar hai, khoj nahi, aur jawab hai ek hazaar aur spins.

Maan-yataayein

Independent spinsganitIndependent spinsHistory kuch bhi ho, har spin ki probabilities same rehti hain. Is site ki har calculation ki buniyaad yahi hai.Poora padhein →, aur har pocket ke kam se kam 5 expected counts, yaani test valid hone ke liye hi n ≥ 185 aur kaam ka hone ke liye kahin zyada. Alag wheels ke spins milana pehli maan-yata ko bhaavna mein todta hai (aap ek mishran test kar rahe hain, wheel nahi). Dealer-signature asar ise asal mein todega, aur bina kisi pocket ke physically favoured hue χ² badha dega.

Roulette Dealer SignatureIndependent Roulette Spins

Pearson aur Monte Carlo

Karl Pearson ne 1900 mein yeh test publish kiya, aur unke pehle hal kiye udaharanon mein se ek 26,306 record kiye gaye Monte Carlo roulette spins the, jinhe unhone chance hone ke liye bahut zyada non-uniform paya aur elaan kiya ki “Monte Carlo roulette… is not a game of chance”. Baad ke lekhakon ne nateeja nikala ki jin journalists ne ek akhbaar ke liye spins record kiye the unhone bahut saare khud gadh liye the, jo data ke source ke baare mein apne aap mein ek seekh hai.

Monte Carlo Roulette

Ek asli test: Las Vegas mein automatic wheels

Players aksar shak karte hain ki automatic wheel ballsaamanBallChhoti ball jo rotor ki ulti disha mein track par chhodi jaati hai. Aaj ki balls ivorine ya Teflon ki hoti hain, 18–21 mm.Poora padhein → ko badi bets se door le jata hai. 2017 ke ek analysis ne Las Vegas mein Interblock ke bina dealer wale double-zerowheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → wheels par is shak ko test kiya, paanch padosi numbers ke arc par bet lagakar, waisi bet jo wheel clocker lagata hai. 299 spins mein arc 39.3 expected ke muqable 29 baar jeeta, aisi kami jo fair wheel lagbhag 4% baar deta hai. Saare 38 pockets par chi-square 37 degrees of freedom par 45.29 tha: p = 16.5%. Teen wheels par 1,204 spins ke doosre sample ne χ² = 34.79, p = 57% diya, aur aazmaya gaya kisi bhi size ka koi arc significant nahi tha. Dono sample milakar, 1,503 spins, ne χ² = 41.75, p = 27% diya. Faisla tha: steering ka koi saboot nahi. Akela pehla sample shakki lagta tha; doosre ne maamla tay kiya.

Chi-square se aage

Ek shakki pocket ke liye binomial test zyada powerful hai. Shakki arc ke liye sector analysis page par sectorwheelSectorWheel par pockets ka lagatar ek hissa (arc), layout ke kisi group se alag cheez.Poora padhein → z-test. Chi-square omnibus test hai: yeh batata hai ki kuch gadbad hai, bina yeh bataye ki kya. Pehle ise chalaiye, phir dekhiye.

Sector Analysis

Formula

Sawaal