Roulette mein Streaks ki Probability
Ek hi color ke lambe silsile aam hain. Agle spin ke liye inka koi matlab bhi nahi hai.
Lagatar paanch red: 2.7%. Das: 0.074%. 200-spin session mein ek color ke chhe ka silsila aadhe se zyada baar dikhta hai.
Haar ki streak kitni baar aati hai
Single-zero wheel par even-money bet 37 mein se 19 spins haarti hai. Har row woh chance hai ki spins ka ek diya hua silsila poora haar jaaye, aur tab tak ek Martingale player, jo har haar ke baad double karta hai, kitna haar chuka hota hai.
| lagatar haar | chance | lagbhag 1 in | ab tak ki haar | agli bet |
|---|---|---|---|---|
| 1 | 51.4% | 2 | $1 | $2 |
| 2 | 26.4% | 4 | $3 | $4 |
| 3 | 13.5% | 7 | $7 | $8 |
| 4 | 7.0% | 14 | $15 | $16 |
| 5 | 3.6% | 28 | $31 | $32 |
| 6 | 1.8% | 55 | $63 | $64 |
| 7 | 0.94% | 106 | $127 | $128 |
| 8 | 0.48% | 207 | $255 | $256 |
| 9 | 0.25% | 403 | $511 | $512 |
| 10 | 0.13% | 784 | $1,023 | $1,024 |
| 11 | 0.065% | 1527 | $2,047 | $2,048 |
| 12 | 0.034% | 2974 | $4,095 | $4,096 |
| 13 | 0.017% | 5792 | $8,191 | $8,192 |
| 14 | 0.0089% | 11279 | $16,383 | $16,384 |
| 15 | 0.0046% | 21965 | $32,767 | $32,768 |
Chance 19/37 ko har haar par ek baar khud se guna karke aata hai, ek diye hue spin se gina hua. Lambe session mein aise kai spins hote hain jahan aisa silsila shuru ho sakta hai, isliye ek shaam mein yeh ek row ke andaaze se kahin zyada mumkin hai.
Ek silsila
Agle k spins sab redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → hone ki probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → (18/37)ᵏ hai. Do: 23.7%. Paanch: 2.7%. Das: 0.074%, lagbhag 1,350 mein 1. Bees: 1.8 million mein 1. Chhabbees, Monte Carlo ka record: 137 million mein 1. Yeh ek khaas spin se shuru hone wale khaas silsile ki probabilities hain, aur yeh sawaal lagbhag koi nahi poochh raha hota.
Ek session mein
Log asal mein yeh poochhte hain ki “kya mere session mein kahin k ka silsila aayega?”, aur iska jawab (18/37)ᵏ se kahin bada hai, kyunki 200-spin session mein silsila shuru hone ki lagbhag 200 jagah hoti hain. Ek hi color, red ya blackbetBlack18 black numbers par even-money bet. French mein: noir.Poora padhein →, ka chhe ya zyada ka silsila 200-spin session mein lagbhag 94% baar aata hai; aath ya zyada lagbhag 47%; das ya zyada lagbhag 14%. Color pehle se tay kar dijiye to chances ghat kar lagbhag 75%, 27% aur 7% ho jaate hain. Jo player poori shaam baithta hai, use har shaam lambe streaks dikhenge.
Martingale ko isse fark kyun padta hai
Har negative progressionstrategyProgressionKoi bhi rule jo pichhle results ke hisaab se stake badalta hai. Negative progressions haar ke baad ise badhati hain, positive jeet ke baad.Poora padhein → ki ek silsile ki lambai hoti hai jo use tod deti hai: $500 limit par $10 base se chhe haar, FibonaccistrategyFibonacciProgression jo sequence 1, 1, 2, 3, 5, 8, 13… follow karti hai: haar ke baad ek kadam aage, jeet ke baad do kadam peeche.Poora padhein → ke liye das, dozensbetDozen1–12, 13–24 ya 25–36, payout 2:1.Poora padhein → par teen guna karne ke liye paanch. Even money par chhe haar ke silsile ki probability har koshish par (19/37)⁶ ≈ 1.8% hai, aur 200-spin session mein shayad 100 koshishein hoti hain, isliye itne lambe sessions mein se lagbhag paanch mein chaar mein yeh toot aa jaata hai. Roulette mein streaks ki probability hi wajah hai ki yeh cliff durlabh nahi hai.
Hot aur cold numbers
Yahi arithmetic straight upsbetStraight-upEk single number, payout 35:1.Poora padhein → par laagu hota hai. 300 spins mein har number ka expected count 8.1 hai; standard deviationganitStandard deviationVariance ka square root; expected value se kisi result ki aam doori.Poora padhein → lagbhag 2.8. Sirf chance se hi koi number 15 hits dikhayega aur koi 2, har baar. “Hot” numbers highlight karne wala results board bas normal distribution ka top highlight kar raha hai. Lab samjhata hai ki hot numbermythHot numberAisa number jo haal mein expected se zyada baar aaya hai. Fair wheel par yeh kuch predict nahi karta.Poora padhein → ka kuch matlab hone ke liye kitne spins chahiye (hazaron, ek pehchaane hue wheel se).
Independence hi asli baat hai
Das reds ka silsila black ko zyada likely nahi banata, aur red ko bhi nahi. Dono 18/37 hain. Streaks random sequences ki khasiyat hain aur unka na hona shak paida karta: 200 spins ka aisa sequence jismein chaar se lamba koi silsila na ho, baarah ke silsile wale sequence se kahin zyada chhed-chhaad ka saboot hota.
Formula
Sawaal
18 August 1913 ko Monte Carlo mein 26 blacks sabse mashhoor hai. 1943 mein ek American casino mein lagatar 32 reds report hue the. Casinos jitne spins chalate hain, uske hisaab se dono chance se mel khaate hain.
Dono taraf 18/37 hai. Jo aapka plan kehta hai woh bet kijiye, ya kuch nahi.
Nahi. Bias hazaron spins mein kuch pocketssaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → ke zyada baar aane se dikhta hai, kisi color ke silsilon se nahi.
Maths ke saath mazaa
Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.
European wheel par average mein kitne spins lagte hain jab tak 37 mein se har number kam se kam ek baar na aa jaaye?
Solution dikhaiye
Jawab Average mein lagbhag 155 spins; median 147 hai.
Intezaar ko ek-ek naye number ke hisaab se giniye. Pehla spin hamesha naya number dikhata hai. 1 number dekh lene ke baad 37 mein 36 pocketssaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → naye hain, isliye agla naya number average mein 37/36 spins leta hai. k numbers dekh lene ke baad yeh 37/(37 − k) leta hai.
Intezaar jodiye: 37/37 + 37/36 + 37/35 + … + 37/1 = 37 × (1 + 1/2 + 1/3 + … + 1/37) ≈ 37 × 4.20 = 155.5 spins.
Iska zyadatar hissa aakhir mein hai. Akela aakhri number average mein 37 spins leta hai, aur aakhri paanch lagbhag 85. American wheel par yahi jod lagbhag 161 deta hai. Yeh coupon collector’s problem hai; yahi mean standard calculation se bhi aata hai.
European wheel ke 37 spins mein, har pocket ke liye ek spin, aapko kitne alag numbers dikhne ki ummeed karni chahiye?
Solution dikhaiye
Jawab Lagbhag 23.6. Kareeb ek-tihaai numbers, lagbhag 13, bilkul nahi aate.
Ek number lijiye, maan lijiye 17. Har spin par uske na aane ka chance 36/37 hai, isliye woh sabhi 37 spins (36/37)³⁷ ≈ 0.363 chance se miss karta hai.
Har number ka chance yahi hai, isliye average mein 37 × 0.363 ≈ 13.4 numbers gayab rehte hain aur 37 − 13.4 ≈ 23.6 aate hain, unmein se kai do ya zyada baar.
Roulette khiladi ise law of thirds kehte hain. Yeh koi law nahi, sirf repeats ka hisaab hai: wheel jitni lambi series mein lagbhag ek-tihaai numbers baahar baithte hain, fair wheel par bhi utna hi jitna kisi aur par. 38 pockets par bhi hissa (37/38)³⁸ ≈ 0.363 hai; bade wheel ke liye yeh 1/e ≈ 0.368 ke paas tik jaata hai.
Ek dost aapko 50 colors, redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → ya blackbetBlack18 black numbers par even-money bet. French mein: noir.Poora padhein →, ki do lists dikhata hai, dono ke baare mein kaha gaya hai ki woh zeroswheelZeroAkela green pocket jise koi even-money, dozen ya column bet cover nahi karta; house edge ka source. Doosre naam: single zero, 0.Poora padhein → chhodkar ek wheel se aayi hain. Ek 25 baar color badalti hai aur uski sabse lambi streak chhe ki hai. Doosri 35 baar badalti hai aur kabhi teen se aage nahi jaati. Kaunsi gadhi gayi hai?
Solution dikhaiye
Jawab Doosri. Asli list lagbhag 24 ya 25 baar badalti hai. 35 badlav wali list 500 lists mein lagbhag ek baar aati hai, aur teen se lambi streak na hone wali lagbhag 100 mein do baar.
Har naya color ya to pichhle ko dohrata hai ya badalta hai, aadha-aadha, isliye badalne ke 49 mauke average mein 24.5 badlav dete hain, lagbhag 3.5 upar ya neeche. Paintees average se aise teen kadam upar hai; fair wheel itne badlav 500 lists mein lagbhag ek baar deta hai.
Streaks bhi yahi kahani kehti hain. 50 ki asli list mein lagbhag 98% baar chaar ya zyada ki streak hoti hai, aur uski sabse lambi streak aksar paanch ya chhe hoti hai.
Jin logon se random list gadhne ko kaha jaata hai woh bahut zyada baar badalte hain, kyunki chaar ya paanch ki streak unhe pattern lagti hai. Taylor (2021) students ki gadhi coin toss lists mein yahi nishaani dikhate hain: asli randomness logon ki ummeed se zyada streaky hoti hai.
Casino kahaniyon ki ek kitaab ek khiladi ki baat karti hai jiska number lagatar paanch baar aaya aur ise record kehti hai. European wheel par yeh kitna rare hai? Aur 20 tables wale ek casino mein, jahan har table din-raat ghante mein 40 baar spin hoti hai, yeh kahin na kahin kitni baar hona chahiye?
Solution dikhaiye
Jawab Ek tay number lagatar paanch baar: 69 million mein 1. Koi bhi number pehli baar aane ke baad chaar aur baar aaye: 1.87 million mein 1. Woh casino saal mein lagbhag 7 million spins karta hai, isliye use saal mein teen-chaar baar paanch ki streak dikhni chahiye.
Ek tay number har spin par 1/37 chance se aata hai, isliye lagatar paanch ka chance (1/37)⁵ = 6,93,43,957 mein 1 hai. Lekin koi pehle se number tay nahi karta. Jo bhi pehle aaye, agle chaar spins use dohraayein: (1/37)⁴ = 18,74,161 mein 1.
20 tables × 40 spins × 24 ghante × 365 din = saal mein 70,08,000 spins. Har spin ek streak shuru kar sakta hai, isliye casino saal mein 70,08,000/18,74,161 ≈ 3.7 paanch wali streaks ki ummeed karta hai.
Is tarah ka koi official record nahi hai, aur zaroorat bhi nahi: ek table par rare ghatna poore casino mein aam baat hai, aur kahaniyan us table ki sunayi jaati hain jahan yeh hua. Goldman (2007) mein kahani double-zero wheel par hai, jahan aankde 79 million mein 1 aur 2.09 million mein 1 hain.
Thodi hansi
Lambe silsile hi random ki asli shakal hain.
Sources
- 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.