Gambler's Fallacy
Yeh maanna ki koi result “due” hai kyunki woh kuch der se nahi aaya.
Das reds ke baad bhi black ki probability 18/37 hai. Wheel koi record nahi rakhta aur kisi ka kuch nahi deta.
Fallacy
Agar redbetRed18 red numbers par even-money bet. French mein: rouge.Poora padhein → das baar aa chuka hai, to kai players ko lagta hai ki blackbetBlack18 black numbers par even-money bet. French mein: noir.Poora padhein → “due” hai aur woh usi hisaab se bet lagate hain. Doosron ko lagta hai red “hot” hai aur woh us par lagate hain. Dono ek khatam ho chuke sequence se ek aise spin ke baare mein soch rahe hain jo abhi hua hi nahi. 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 nahi pata ki pichhle das spins kya the; pocketssaamanPocketRotor ke 37 numbered khaanon mein se ek. Doosre naam: slot, cup.Poora padhein → utne hi bade hain jitne ek ghante pehle the; black ki probabilityganitProbabilityKisi outcome ka chance, 0 se 1 tak ya 0% se 100% tak. Ek number: 1/37 = 2.70%.Poora padhein → hamesha ki tarah 18/37 hai.
Yeh sach kyun lagta hai
Lambe samay mein red aur black ka anupaat sach mein har ek ke liye 18/37 ki taraf jaata hai. Law of large numbersganitLaw of large numbersTrials ki ginti badhne par independent results ka average expected value ki taraf jaata hai.Poora padhein → iski guarantee deta hai. Lekin yeh dilution se hota hai, correction se nahi: 1,000 aur spins ke baad das extra reds anupaat mein bas rounding error hain, isliye nahi ki black ne barabari kar li, balki isliye ki sample bada ho gaya. Black ko zyada likely banane wala koi mechanism nahi hai, aur law of large numbers ko aisa koi chahiye bhi nahi.
Monte Carlo, 18 August 1913
Casino de Monte-Carlo ki ek table par black lagatar 26 baar aaya. Lagbhag pandrahvein black se players red par toot pade, double aur phir double karte hue, har spin pichhle se “zyada pakka”. Casino ne laakhon francs jeete. Pehle black se is silsile ki probability 137 million mein 1 thi, aur duniya ke casinos mein kahin na kahin aakhirkar iske hone ki probability lagbhag ek thi. Isi wajah se gambler’s fallacy ko Monte Carlo fallacy bhi kehte hain.
Kuch khaas nahi. 44 mein ek baar hota hai.
Players dhyaan dene lagte hain. Red “due” hai.
Table par bheed hai. Har haar ke baad red par stakes double kiye jaate hain.
Daulat lut gayi. Har naya black red ko aur pakka mehsoos karata hai.
Silsila khatam. Casino ne un logon se laakhon francs jeete jo ek aise silsile ke khilaaf bet laga rahe the jo “jaan” hi nahi sakta tha ki woh silsila hai.
Hot hand
Kisi color ko isliye back karna kyunki uska silsila chal raha hai, wahi galti ulti karke hai: ummeed ki jaati hai ki silsila khatam hone ke bajaye chalta rahega. Psychologists ise hot handmythHot handYeh maanna ki jeet ka silsila, ya ek hi result ka silsila, agli baar wahi hone ko zyada likely bana deta hai. Wheel par yeh ulta gambler’s fallacy hai.Poora padhein → kehte hain, Gilovich, Vallone aur Tversky ki 1985 ki study ke naam par, jisne basketball shooting mein streaks dhoondhe aur chance se zyada kuch nahi paaya. 2018 mein Miller aur Sanjurjo ne dikhaya ki test khud jhuka hua tha: chhote record mein hit ke theek baad hits ka hissa average mein shooter ke overall rate se neeche aata hai, isliye bina hot hand wala player thoda cold dikhta hai. Sudhaarne par wahi data ek halka hot hand dikhata hai. Insaan garm ho sakta hai; wheel ki koi form hi nahi hoti. Kuch players dono baatein ek saath maante hain, ki silsila tab tak chalta hai jab tak “bahut lamba” na ho jaaye aur phir khatam hota hai. Fair wheel par har version ki expected valueganitExpected valueLambe samay mein har bet ka ausat nateeja. Har standard European bet ke liye stake ka −1/37.Poora padhein → coin uchhaal kar color chunne jitni hai: stake ka −2.70%.
Asli table par
Croson aur Sundali (2005) ne Reno ke ek bade casino ki ek 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 → table ka 18 ghante ka security video dekha: 139 players, 24,131 bets. Even-money betsbetEven-money betRed/black, odd/even, high/low. 18 numbers, payout 1:1.Poora padhein → par players chhote silsilon ko follow karne aur unke khilaaf jaane mein lagbhag barabar bante the. Lagatar paanch ek jaise ke baad woh silsile ke khilaaf jhuke, aur chhe ya zyada ke baad unke 85% bets doosri taraf the. Wahi players apne baare mein hot hand dikhate the: numbers par jeet ke baad agle spin par average mein 13.6 numbers cover karte the, jabki haar ke baad 9.2. Unke wheel mein bias ka koi nishaan nahi tha, isliye isse kisi ki expectation nahi badli.
Casino ke bahar
Yahi khinchaav wahan bhi dikhta hai jahan asli faisle hote hain. Clotfelter aur Cook (1993) ne Maryland ka daily numbers game follow kiya: kisi number par lagaya paisa uske draw hote hi tezi se gira aur wapas aane mein mahine lage, jabki draw ki koi yaaddasht nahi hoti. Chen, Moskowitz aur Shue (2016) ne ise professionals mein paaya. United States ke asylum judges pichhla case manzoor karne ke baad agla case thukraane ki 3.3 percentage points tak zyada sambhavna rakhte the. Baseball umpires ek strike dene ke baad agla strike dene ki 1.5 points kam sambhavna rakhte the, aur kareebi pitches par iske do guna se zyada. Flat fee paane wale loan officers ke ek field experiment mein 9% tak faisle is tarah palat gaye. Cases random order mein aate the, isliye pichhla faisla agle ke baare mein kuch nahi batata tha, theek jaise pichhla spin agle ke baare mein kuch nahi batata.
Inverse gambler’s fallacy
Philosopher Ian Hacking (1987) ne ek teesre version ka naam rakha. Koi kamre mein aata hai, dekhta hai ki dice ka joda double six par gira, aur nateeja nikaalta hai ki yeh pehle bhi kai baar phenke ja chuke honge. Har roll ke double six hone ka chance barabar hai, shaam ka pehla ho ya hazaarvan, isliye yeh nazaara kuch nahi batata ki game kab se chal raha hai. Roulette mein yeh woh player hai jo aath reds dikhate board par pahunchta hai aur maan leta hai ki inhe banane ke liye table ghanton se chal rahi hogi. Aam fallacy ateet ko agle spin mein padhti hai; yeh ek durlabh nazaare ko aise ateet mein padhti hai jise kisi ne dekha nahi.
Ek psychologist jo ise maanta tha
Is fallacy ke padhe-likhe himayati bhi rahe hain. German psychologist Karl Marbe ne 49,152 births wale chaar registers dekhe, paaya ki lagatar ladkon ya ladkiyon ka sabse lamba silsila 17 tha, aur nateeja nikaala ki lambe silsile chance se kam hote hain, isliye silsila doosre outcome ko zyada likely banata hai. Akela chance kuch aur kehta hai: 1,96,608 births mein 17 ya zyada ka silsila lagbhag paanch mein chaar baar aata hai. Marbe ne theek wahi paaya tha jo random sequence jaisa dikhta hai. Epstein (2013) yeh case dobara sunate hain; Probability of Streaks iska roulette version deta hai.
Jahan fallacy laagu nahi hoti
Agar wheel physically biased hota, to pichhle spins kaam ki jaankari dete, kyunki woh kismat ke bajaye pockets ke baare mein saboot hote. Yeh Lab ka vishay hai, aur kuch kehne ke liye ek pehchaane hue wheel se hazaron spins chahiye. Anjaan wheel par das reds ke silsile ke liye bias wali vyakhya chance wali se kahin kam likely hai.
Sawaal
Nahi. Yeh kisi bhi independent dohraaye jaane wale event par laagu hoti hai: dice, coins, lottery numbers.
Anupaat kareeb aa jaayenge. Aapka result phir bhi aapke wager ka lagbhag −2.70% hoga.
Law of averages jaisa kuch nahi hai. Law of large numbers hai, aur woh dilution se kaam karta hai.
Maths ke saath mazaa
Pehle khud try kijiye. Har solution chhupa hua hai; jab taiyaar hon tab kholiye.
10 tosses ke baad ek coin 7 heads aur 3 tails dikhata hai. 1,000 tosses ke baad 517 heads aur 483 tails. Kya tails ne barabari kar li?
Solution dikhaiye
Jawab Nahi. Fark 4 se badhkar 34 ho gaya. Sirf ratio barabari ke kareeb aaya.
Heads aur tails ka ratio 7/3, lagbhag 2.33, se girkar 517/483, lagbhag 1.07 ho gaya. Yeh law of large numbersganitLaw of large numbersTrials ki ginti badhne par independent results ka average expected value ki taraf jaata hai.Poora padhein → ka kaam hai.
Dono ke beech ka fark badha, 4 se 34. Fair coin ke saath 1,000 tosses ke baad lagbhag 30 ka fark aam baat hai: typical fark mote taur par tosses ki ginti ke square root ke saath badhta hai, jabki ratio 1 ki taraf tikta hai.
Kisi cheez ne tails ko wapas nahi kheencha. Heads ki shuruaati badhat agle 990 tosses mein ghul gayi, theek nahi ki gayi.
Ek coin chaar baar uchhaliye aur sirf un tosses ko dekhiye jo head ke theek baad aate hain. Unmein se kitna hissa heads hai? 16 sambhav sequences mein se har ek ke liye nikaliye aur hisson ka average lijiye. Kya yeh 50% hai?
Solution dikhaiye
Jawab Nahi. Yeh 17/42 hai, lagbhag 40.5%. Is tarah dekhne par fair coin head ke baad thanda padta lagta hai.
16 sequences ki list banaiye. Unmein se do, TTTT aur TTTH, ke pehle teen tosses mein koi head nahi, isliye head ke baad kuch nahi aata; 14 bachte hain.
Har ek mein, head ke baad aane wale tosses lijiye aur dekhiye unmein kitna hissa heads hai. HHHH 3 mein se 3 deta hai. HHHT 3 mein se 2. HTHT 2 mein se 0. Saare 14 dekhne par: chhah sequences 0 dete hain, chaar 1/2, ek 2/3 aur teen 1 (HHHH, THHH aur TTHH).
Hisse jodiye: 4 × 1/2 + 2/3 + 3 = 17/3. 14 se bhaag dijiye: 17/42 ≈ 40.5%.
Koi toss tails ka favour nahi karta; har ek 50–50 hai. Jhukaav sequence dar sequence average lene se aata hai: heads se bhare sequence mein dekhne ke liye kai tosses hote hain lekin woh ek hi baar gina jaata hai, ek single head wale sequence jitna. Miller aur Sanjurjo (2018) ne dikhaya ki 1985 ki hot-hand study ne apne basketball players ko isi tarah naapa tha, isliye jo players chance se zyada streaky nahi lagte the, woh asal mein kuch zyada streaky the.
Thodi hansi
Dilution, correction nahi. Wheel kisi ka kuch nahi deta.
Sources
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Schwartz, D. G. Roll the Bones: The History of Gambling.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Croson, R. & Sundali, J. “The gambler’s fallacy and the hot hand: empirical data from casinos.”
- Chen, D. L., Moskowitz, T. J. & Shue, K. “Decision making under the gambler’s fallacy: evidence from asylum judges, loan officers, and baseball umpires.”
- Clotfelter, C. T. & Cook, P. J. “The ‘gambler’s fallacy’ in lottery play.”
- Gilovich, T., Vallone, R. & Tversky, A. “The hot hand in basketball: on the misperception of random sequences.”
- Miller, J. B. & Sanjurjo, A. “Surprised by the hot hand fallacy? A truth in the law of small numbers.”
- Hacking, I. “The inverse gambler’s fallacy: the argument from design. The anthropic principle applied to Wheeler universes.”