Sawaal: Birthday Wheel
Kitne spins ke baad koi number dobara aata hai? Zyadatar logon ke andaaze se kahin kam.
37-pocketsaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → wheel par birthday problem: sirf 8 spins ke baad aadhe se zyada chance hai ki koi number pehle hi do baar aa chuka hai. Players repeats ko pattern samajhte hain; yeh fair wheel ka expected vyavhaar hai.
Sawaal
Ek fair European wheel ghumaaiye. Kitne spins tak probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → ki kam se kam do spins par ek hi number aaya ho, aadhe se zyada ho jaati hai? Zyadatar players 18 ya 19 ke aas-paas andaaza lagaate hain, 37 ka aadha. Jawab 8 hai.
Hal
Koi repeat na hone ki probability nikaaliye aur ek mein se ghataaiye. k spins ke baad, p(sab alag) = (37/37)(36/37)(35/37)…((38 − k)/37). k = 8 ke liye: 1 × 0.973 × 0.946 × 0.919 × 0.892 × 0.865 × 0.838 × 0.811 ≈ 0.44, isliye p(koi repeat) ≈ 0.56. k = 14 ke liye yeh 0.93 hai; k = 20 ke liye 0.996. Intuition isliye fail hota hai kyunki spins ki jodiyon ki ginti k² ki tarah badhti hai, aur koi bhi jodi match ho sakti hai.
Table par yeh kyun maayne rakhta hai
Pichhle 20 numbers ka results board lagbhag hamesha kam se kam ek repeat dikhayega, aksar do ya teen. Players dekhte hain “17 das spins mein do baar aaya” aur maan lete hain ki 17 hot hai; birthday arithmetic kehta hai ki das spins mein repeat 74% baar hota hai aur repeat hone wala number random hai. Pattern asli hai; uska matlab nahi.
Agla sawaal
Kitne spins tak koi khaas number, maan lijiye 17, do baar aa chuka hoga? Yeh alag sawaal hai jiska jawab kahin bada hai: expected intezaar 74 spins hai, aur 20 spins mein do 17 aane ki probability lagbhag 10% hai. “Koi number repeat hota hai” (8 spins) aur “yeh number repeat hota hai” (74 spins) ka fark hi baad mein dikhne wale pattern aur pehle se ki gayi prediction ka poora fark hai.
Khud try kijiye
Demo Game das spins ke liye chalaaiye aur history dekhiye. Aam taur par ek repeat hoga. Phir chalaaiye. Repeat hone wala number alag hoga. Aisa das baar kijiye aur aapne birthday problemganitBirthday problemKitne logon ke baad do logon ka birthday shayad same hoga, ya kitne spins ke baad koi number shayad repeat hoga. 37-pocket wheel par 8 spins hi aadhe se zyada chance ke liye kaafi hain.Poora padhein → dekh liya; hazaar baar kijiye (simulators kar sakte hain) aur aapne use maap liya.
Formula
Sawaal
Bilkul, bas 365 dinon ki jagah 37 pockets. Birthdays ke liye 23 log 50% dete hain; wheel ke liye 8 spins 56% dete hain.
Lagbhag 19 spins ke baad 50%. Phir bhi aam baat.
Nahi. Ek wheel ke hazaron spins ho sakte hain; ek repeat kabhi nahi.
Maths ke saath mazaa
Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.
American wheel ke bagal ka results board pichhle 16 spins dikhata hai. Kitna sambhav hai ki us par koi number do baar na dikhe?
Solution dikhaiye
Jawab Lagbhag 2.5%: mote taur par 40 mein se 39 boards par kam se kam ek repeat dikhta hai.
Spin dar spin chaliye. Pehla spin kuch bhi ho sakta hai. Doosre ko use chookna hai: 38 mein 37 chances. Teesre ko dono ko chookna hai: 38 mein 36. Solahve ko pandrah numbers chookne hain: 38 mein 23.
Guna kijiye: 38/38 × 37/38 × 36/38 × … × 23/38 ≈ 0.0246.
Bina repeat wala board hi durlabh hai. Repeat wala board wheel ke baare mein kuch nahi batata.
European wheel ke das spins mein, 17 ke theek paanch baar aane ka chance kya hai? Kisi bhi ek number ke paanch ya zyada baar aane ka? Aur chaar spins mein kisi number ke kam se kam teen baar aane ka?
Solution dikhaiye
Jawab 17 theek paanch baar: 3,15,576 mein 1. Koi number paanch ya zyada baar: lagbhag 8,333 mein 1. Chaar spins mein koi number teen ya zyada baar: 349 mein 1.
17 paanch baar: das spins mein se kaun se paanch, yeh chuniye, 252 tareeke, har ek ka chance (1/37)⁵ × (36/37)⁵. Kul milakar, 3,15,576 mein 1.
Koi bhi number: 37 numbers mein se har ek ke paanch ya zyada hits ka chance 3,08,325 mein 1 hai, jo 3,08,325 mein 37, yaani 8,333 mein 1 deta hai. Do numbers ke paanch-paanch hits wala durlabh case do baar gina jaata hai, lekin use hatane se is precision par kuch nahi badalta.
Chaar spins mein teen baar: ek number ke liye, 37⁴ sambhav results mein se 4 × 36 + 1 = 145; 37 mein se kisi bhi number ke liye, 145/37³ = 349 mein 1.
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 → wheel ke liye yahi figures 3,59,275 mein 1, 9,243 mein 1 aur 368 mein 1 hain. “17” aur “koi bhi number” ke beech ka fark phir se birthday problemganitBirthday problemKitne logon ke baad do logon ka birthday shayad same hoga, ya kitne spins ke baad koi number shayad repeat hoga. 37-pocket wheel par 8 spins hi aadhe se zyada chance ke liye kaafi hain.Poora padhein → hai: hairan hone ke 37 tareeke hain.
Thodi hansi
Kuch na kuch jaldi repeat hota hai. Woh kuch random hai.
Sources
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Wizard of Odds. “Ask the Wizard”, roulette questions.