LABEXPERIMENT 06df = 36α = 0.05seed 99

Roulette Wheel Bias

Biased roulette wheel kisi physical kharabi ki wajah se kuch pockets ko favour karta hai: dheela fret, asamaan rotor, ghise pockets. Roulette wheel bias hota hai aur iska faayda uthaya gaya hai. Mushkil yeh hai ki bilkul fair wheel bhi lagatar “hot numbers” paida karta hai, isliye sawaal kabhi yeh nahi ki koi number hot hai, balki yeh ki poora pattern chance ki ijaazat se zyada tedha hai ya nahi.

1. Fair wheel ka sample

Bina kisi bias wale simulated wheel se nikale gaye spins. Sample size badaliye aur sabse garam pocket dekhiye.

spins370

har pocket par expected10.0

sabse garam pocket20 · 16 hits (4.32%)

sabse garam vs expected+60% · 1.9σ

chi-square · p32.8 · 0.622

faislabias ka koi saboot nahi

2. Observed banaam expected

Har bar wheel order mein ek pocket hai. Dashed line expected count, n/37, hai. Shaded band ±2 standard deviations hai: fair wheel par sirf chance se lagbhag bees mein ek pocket iske bahar niklega.

0wheel order mein pockets →26

3. Chi-square test

Test jodta hai ki har pocket expectation se kitna door hai, square aur scale karke. Fair wheel par jod 36 degrees of freedom wale chi-square distribution ko follow karta hai; 0.05 se kam p-value ko parampara se significant kaha jata hai.

χ² = Σ (Oᵢ − Eᵢ)² / Eᵢ Eᵢ = n/37 df = 36
n = 370 · E = 10.00 · χ² = 32.80 · p = 0.622

Sample-size ki chetavniLagbhag 1,000 spins se neeche test mein realistic biases pakadne ki lagbhag koi power nahi hoti, aur p < 0.05 par “significant” nateeja aapke dekhe har 20 fair samples mein 1 mein chance se aayega. Bahut saare wheels test karna, ya ek hi wheel ko paas hone tak dobara test karna, false positives pakke karta hai.

4. Aapko kitne spins chahiye

5% one-sided level aur 80% power par 1/37 se zyada lagne wale ek pocket ko pakadne ke liye zaroori spins. 37 mein 1 ki jagah 35 mein 1 baar lagne wala pocket player ke liye mazboot edge hoga, aur uski pushti mein das-hazaaron record kiye spins lagte hain.

asli rateplayer edgezaroori spins40 spins/h par

25 mein 1+44.0%1,10828 h

30 mein 1+20.0%4,392110 h

33 mein 1+9.1%15,747394 h

35 mein 1+2.9%69,4611,737 h

Bias ki pushti ka matlab hai raat dar raat spins record karna, das-hazaaron.

Methodology

Sampling. Seeded generator (mulberry32, seed 99) se 37 pockets mein se uniform draws, taaki page reproducible rahe.

Test. Pearson chi-square goodness of fit, 36 degrees of freedom, regularized incomplete gamma function se p-value.

Power. One-proportion test ka normal approximation, α = 0.05 one-sided, power 0.80.

Daayra. Sirf statistical detection. Physical kaaran aur sector-based (pocket-based ki jagah) analysis Sector Analysis mein hain.

Thodi hansi

Fair wheel har raat hot numbers banata hai. Bias saabit karne mein das-hazaaron spins lagte hain.

Sources

Pearson, K. (1900). On the criterion that a given system of deviations… Philosophical Magazine, 50(302).
Thorp, E. O. (1969). Optimal gambling systems for favorable games. Review of the International Statistical Institute, 37(3).
Ethier, S. N. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer. Ch. 13, Roulette.