16.3: The Gambler’s Fallacy
- Page ID
- 95161
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We commit the gambler’s fallacy when we treat things that are independent as though they were not independent. In other words, when we (mistakenly) think that one of two independent things influence the other. For example, the outcomes of successive flips of a fair coin are independent of each other, so the outcome of the second flip does not depend in the least on the outcome of previous flips. If you flip a fair coin ten times and it comes up heads each time, the probability of it coming up heads on the eleventh flip is still 1/2.
Of course, if you get enough heads in a row you may begin (quite reasonably) to suspect that the coin really isn’t fair. But even if it is biased, so that it is likely to come up heads twice as often as tails, the point remains: the outcomes of the two successive flips are independent of each other, so what happens on the next flip isn’t affected by earlier outcomes.
In such situations, we tend to think that the coin is more likely to come up tails in order to “even things out,” to satisfy the “law of averages.” But the coin doesn’t “remember” what it did on earlier flips, and people who reason this way commit the gambler’s fallacy. Similarly, defective thinking is common with other games of chance like roulette, and it is a danger in any reasoning involving probabilities.
The gambler’s fallacy is not restricted to games of chance. Suppose that Wilbur and Wilma have four children, all boys. They would like to have a girl, and they reason as follows. Very nearly half of the children born in the world are girls. We have had four boys in a row, so it’s got to be time that we get a girl. It is an empirical question whether having a child of one sex affects the probability of the sex of subsequent children. The evidence strongly suggests that it does not; the sex of one child is independent of the sex of its siblings. So, assuming that the gender of a couple’s children are independent of one another, Wilbur and Wilma commit the gambler’s fallacy.
There is a saying that lightning never strikes in the same place twice, and some people will even seek refuge in spot where lightning struck before in hopes of being safe. It may be true that lightning rarely strikes in the same place twice, but that is simply because the probability of it striking in any specific spot is reasonably low. But the lightning now doesn’t know where lightning has struck before, and this general slogan “never in the same place twice” rests on the gambler’s fallacy.
Nothing in these cases requires us to have any precise ideas about probability values. If we have good reason to think that two things are independent, we shouldn’t act as though one could influence another. For example, we may have reason to think that Wilbur’s die is loaded so that sixes are more likely to come up than any other number. We may not know how much more likely sixes are, but if the outcomes of separate throws are independent, only bad reasoning can lead us to suppose that since a six hasn’t come up the last ten throws, a six must be due on the next throw.