Gambler Fallacy
Di belief sey one result “don due” becos e never show for some time.
Afta ten reds, di probability of black still be 18/37. Di wheel no dey keep record and e no owe anybody anytin.
Di Fallacy
If redbetRedEven-money bet on di 18 red numbers. For French: rouge.Read di whole tin → don show ten times, many players dey feel sey blackbetBlackEven-money bet for di 18 black numbers. For French: noir.Read di whole tin → “don due” and dem go bet am. Oda ones dey feel sey red dey “hot” and dem go bet dat one. Both of dem dey reason from sequence wey don already finish to spin wey never happen. Di ballequipmentBallDi small ball wey dem dey throw for di track against di rotor. Modern balls na ivorine or Teflon, 18–21 mm.Read di whole tin → no sabi wetin di last ten spins be; di pocketsequipmentPocketOne of di 37 small boxes wey get number for di rotor. Dem also dey call am: slot, cup.Read di whole tin → still get di same size wey dem get one hour ago; di probabilitymathsProbabilityDi chance of one outcome, from 0 to 1 or from 0% to 100%. One number: 1/37 = 2.70%.Read di whole tin → of black na 18/37 as e always be.
Why E Be Like Sey E True
For di long run, di proportions of red and black dey really come close to 18/37 each. Di law of large numbersmathsLaw of large numbersDi average of results wey no depend on each oda dey move near di expected value as di number of tries dey grow.Read di whole tin → guarantee am. But di convergence dey happen by dilution, no be by correction: afta 1,000 more spins, di ten extra reds go be rounding error for di proportion, no be becos black catch up but becos di sample grow. No mechanism dey wey dey make black more likely, and di law of large numbers no need one.
Monte Carlo, 18 August 1913
Black show 26 times one afta di oda for one table for di Casino de Monte-Carlo. From about di fifteenth black, players pile dia moni on red, dem dey double and double again, every spin “more sure” pass di last one. Di casino win millions of francs. From di first black, di run get probability of 1 out of 137 million, and di probability sey e go happen one day somewhere for di casinos of di world na almost one. Na di reason why dem also dey call di gambler fallacy di Monte Carlo fallacy.
Notin strange. E dey happen one time out of 44.
Players begin notice. Red “don due”.
Di table don full. Dem dey double di stakes on red afta every loss.
Big fortunes don lose. Every new black dey make red feel more sure.
Di run end. Di casino don win millions of francs from pipo wey dey bet against one run wey no fit “sabi” sey e be run.
Di Hot Hand
To back one colour becos e dey for run na di same mistake wey dem turn around: you dey expect di run to continue instead of to end. Psychologists dey call am di hot handmythHot handDi belief sey wins wey follow each oda, or one result wey dey repeat, go make anoda one more likely. For di wheel, na di gambler fallacy wey dem turn around.Read di whole tin →, afta one 1985 study by Gilovich, Vallone and Tversky wey look for streaks for basketball shooting and no find any one wey pass chance. For 2018, Miller and Sanjurjo show sey di test imself don bend: for short record, di share of hits wey come immediately afta hit dey come out below di overall rate of di shooter for average, so player wey no get hot hand at all go look small cold. When dem correct am, di same data show small hot hand. Person fit warm up; wheel no get form to dey inside. Some players dey hold both beliefs at once, sey run go continue until e “too long” and then e go end. For fair wheel, every version get di same expected valuemathsExpected valueDi average result for each bet for di long run. −1/37 of di stake for every standard European bet.Read di whole tin → as to choose colour by tossing coin: −2.70% of di stake.
For Real Table
Croson and Sundali (2005) watch 18 hours of security video from one double-zerowheelZeroDi one green pocket wey no even-money, dozen or column bet dey cover; na im be di source of di house edge. Dem also dey call am: single zero, 0.Read di whole tin → table for one big casino for Reno: 139 players wey put 24,131 bets. For di even-money betsbetEven-money betRed/black, odd/even, high/low. 18 numbers, e dey pay 1:1.Read di whole tin →, di players divide almost equal between di ones wey dey follow short runs and di ones wey dey go against dem. Afta five of di same kind one afta di oda, dem lean against di run, and afta six or more, 85% of dia bets dey for di oda side. Di same players show di hot hand about demself: afta win for di numbers, dem cover 13.6 numbers for di next spin for average, compared to 9.2 afta loss. Dia wheel no show any sign of bias, so no one of dis change wetin any of dem fit expect.
Outside Di Casino
Di same pull dey show where pipo dey make real decisions. Clotfelter and Cook (1993) follow di daily numbers game of Maryland: di moni wey pipo stake on one number drop well well as soon as dem draw am, and e take months to recover, even though di draw no get memory. Chen, Moskowitz and Shue (2016) find am among professionals. Asylum judges for United States dey up to 3.3 percentage points more likely to refuse one case if dem grant di one before am. Baseball umpires dey 1.5 points less likely to call strike afta dem call one, and pass double dat one for close pitches. For one field experiment with loan officers wey dem pay flat fee, up to 9% of decisions turn by dis way. Di cases come for random order, so di last decision no tok anytin about di next one, just as di last spin no tok anytin about di next one.
Di Inverse Gambler Fallacy
Di philosopher Ian Hacking (1987) name one third version. Somebody waka enter one room, see one pair of dice land double six, and conclude sey dem must don roll am many times before. Every roll get di same chance to be double six, di first one of di evening or di thousandth, so wetin e see no tok anytin about how long di game don dey go. For roulette, na di player wey reach one board wey show eight reds and decide sey di table must don dey spin for hours to produce dem. Di normal fallacy dey read di past into di next spin; dis one dey read one rare sight into past wey nobody see.
Psychologist Wey Believe Am
Di fallacy don get learned defenders. Di German psychologist Karl Marbe go through four registers of 49,152 births each, find sey di longest run of boys or of girls one afta di oda na 17, and conclude sey long runs rare pass wetin chance allow, so run dey make di oda outcome more likely. Chance alone tok different tin: for 196,608 births, run of 17 or more dey show about four times out of five. Marbe find exactly how random sequence dey look. Epstein (2013) tell di case again; Probability of Streaks for Roulette dey give di roulette version.
Where Di Fallacy No Apply
If di wheel get physical bias, past spins go carry information, becos dem go be evidence about di pockets instead of about luck. Na dat one be di subject of di Lab, and e need thousands of spins from one wheel wey you identify before e fit tok anytin. For run of ten reds for wheel wey you no sabi, di bias explanation no likely like di chance one at all.
Questions wey pipo dey ask
No. E dey apply to any independent event wey dey repeat: dice, coins, lottery numbers.
Di proportions go come close. Your result go still be about −2.70% of wetin you bet.
No law of averages dey. Na law of large numbers dey, and e dey work by dilution.
Play with Maths
Try dem by yourself first. Each solution dey folded; open am when you ready.
Afta 10 tosses, one coin show 7 heads and 3 tails. Afta 1,000 tosses, e show 517 heads and 483 tails. Di tails catch up?
Show di solution
Answer No. Di gap grow from 4 reach 34. Na only di ratio come near even.
Di ratio of heads to tails fall from 7/3, about 2.33, to 517/483, about 1.07. Na di law of large numbersmathsLaw of large numbersDi average of results wey no depend on each oda dey move near di expected value as di number of tries dey grow.Read di whole tin → dey work be dat.
Di gap between dem grow, from 4 reach 34. With fair coin, gap of around 30 afta 1,000 tosses na normal tin: di usual gap dey grow roughly with di square root of di number of tosses, while di ratio dey settle go 1.
Notin pull di tails back. Di next 990 tosses dilute di extra heads wey come early, dem no correct am.
Toss one coin four times and look only di tosses wey come right afta head. Which share of dem na heads? Work am out for each of di 16 sequences wey fit happen and take di average of di shares. E be 50%?
Show di solution
Answer No. Na 17/42, about 40.5%. If you look am dis way, fair coin go be like sey e dey cold afta head.
List di 16 sequences. Two of dem, TTTT and TTTH, no get head among di first three tosses, so notin dey follow head; dat one leave 14.
For each one, take di tosses wey follow head and find di share wey be heads. HHHH dey give 3 out of 3. HHHT dey give 2 out of 3. HTHT dey give 0 out of 2. If you go through all 14: six sequences dey give 0, four dey give 1/2, one dey give 2/3 and three dey give 1 (HHHH, THHH and TTHH).
Add di shares: 4 × 1/2 + 2/3 + 3 = 17/3. Divide by 14: 17/42 ≈ 40.5%.
No toss dey favour tails; each one na 50–50. Di tilt dey come from di way you average sequence by sequence: sequence wey full of heads get plenty tosses to look but e dey count once, di same as sequence wey get only one. Miller and Sanjurjo (2018) show sey di hot-hand study of 1985 measure im basketball players dis way, so players wey be like sey dem no get more streak pass wetin chance dey give, really get small more streak pass chance.
Small laugh
Dilution, no be correction. Di wheel no owe anybody.
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.”