Gambler's fallacy

The gambler's fallacyonemany common misunderstandings which ariseeveryday reasoning about probabilities, manywhich have been studied ingreat detail. Itconsideredbelogical fallacy.

The gambler's fallacy can be illustrated bygamewhichcointossed overover again. Suppose thatcoinin fact fair, so thatchancesit coming up headsexactly 0.5 (a half). Thenchancesit coming up heads twicesuccession0.5×0.5=0.25 (a quarter); three timessuccession, they0.125 (an eighth)so on. In general,'runs' consistingan unbroken successionheadsmoremore unlikely.

Nothing fallacious so far; but suppose that wein onethese states where, say, four heads have just come up inrow,someone argues as follows:runfive successive headsvery unlikely (the probabilityin fact one-thirtysecond, or 0.03125), so itmore likely thatnext toss will betail thanhead. That isfallacy:idea thatrunluck inpast somehow influencesodds ofbet infuture. Related fallacious ideasinherentsuch phrases as "a lucky streak" or "a winning streak" or"break".

Sinceodds ofrunfive headsindeed very low, one might wonder wherefallacy lies. The pointthat those oddslow given no prior information. But atpoint whenfallacyformulated, fourthose headsalready tossed; thereno uncertainty about them at all. Given that wealready instate which itself hadprobabilityonly one-sixteenth (the oddsgetting four heads inrow), we can be sure thatnext state willfact have an uncertaintyone-thirtysecond no matter whatnext toss is. The odds ofheads onnext toss (forfair coin)even, no matter whatpast history ofgamble has been.

Sometimes, gamblers argue like this: "I just lost four times. Sincecoinfairtherefore inlong run everything haseven out, if I just keep playing, I will eventually win my money back." The probabilityindeed equalone thatgambler will eventually win his money back; however,expected numbertimes he hasplayinfinite,so isexpected amountcapital he will need! A similar argument shows thatpopular doubling strategy (start$1, if you lose, bet $2, then $4 etc., until you win) does not work. Situations like theseinvestigated inmathematical theoryrandom walks.

Notice thatgambler's fallacyquite different fromfollowing pathreasoning (which comes toopposite conclusion):coin comes up heads more often than tails, so itnotfair coin, so I will bet thatnext toss will be heads also. Thisnot fallacious, thoughfirst step -argument fromfinite numberobservations tostatementlikelihood - isvery delicate matter,is itself pronefallaciesits own peculiar kind.