Jul 21 2008
Examples of the gambler’s fallacy
A more subtle version of the fallacy is that an “interesting” (non-random looking) outcome is “unlikely” (for instance, that a sequence of “1,2,3,4,5,6″ in a lottery result is less likely than any other individual outcome). Even apart from the debate about what constitutes an “interesting” result, this can be seen as a version of the gambler’s fallacy because it is saying that a random event is less likely to occur if the result, taken in conjunction with recent events, will produce an “interesting” pattern.
As an example, the popular doubling strategy of the Martingale betting system (where a gambler starts with a bet of $1, and doubles their stake after each loss, until they win) is flawed. Situations like these are investigated in the mathematical theory of random walks. This and similar strategies either trade many small wins for a few huge losses (as in this case) or vice versa. With an infinite amount of working capital, one would come out ahead using this strategy; as it stands, one is better off betting a constant amount if only because it makes it easier to estimate how much one stands to lose in an hour or day of play.
In fact a fair gambling system is one on which the probability of some outcome remains exactly the same on each occasion: a roulette wheel has no memory.
It should be noticed that if a system is not known to be like this there may be no fallacy. If we are betting on the weather, or a horse, it may be quite reasonable to take a sequence of rainy days, or a sequence of wins, as increasing the chance of another.
Other examples
What is the probability of flipping 21 heads in a row, with a fair coin?
Answer: 1 in 2,097,152 = approximately 0.000000477.
What is the probability of doing it, given that you have already flipped 20 heads in a row?
Answer: 0.5.
Will you eventually come out ahead at roulette by betting double what you lost the previous time, and adding an extra amount? (Answer: given infinite time and funds, yes, you will eventually win on that color in a fair game. However, given finite time and even more finite funds, the chance exists that you will exhaust your money before winning. Regardless of the odds of a color losing (or winning) several times in a row, the probability of the ball landing on that color in a given spin is the number of that color that exist, divided by all possibilities. In the case of a Vegas roulette wheel, the chances of hitting red are 18/38, or ~.47, regardless of previous results.)
Are you more likely to win the lottery jackpot by choosing the same numbers every time or by choosing different numbers every time? (Answer: Either strategy is equally likely to win.)
Are you more or less likely to win the lottery jackpot by picking the numbers which won last week, or picking numbers at random? (Answer: Either strategy is equally likely to win, but if others choose the same numbers your payout is likely to be less.
A rational gambler might attempt to predict other players’ choices and then deliberately avoid these numbers.
