MegaMillions and the Gambler’s fallacy

What better time to discuss the gambler’s fallacy than tonight 150 minutes or so before MegaMillions lets us know which set of 5 numbers will win over a billion dollars (before taxes).

People generally fail to produce random sequences by overusing alternating patterns and avoiding repeating ones — the gambler’s fallacy that repeating sequences are rare (hence improbable)

Although the gambler’s fallacy is just that, a fallacy, it has a neural basis in terms of the way we think [ Proc. Natl. Acad. Sci. vol. 112 pp. 3788 – 3792 ’23 ].

Here’s why.  There is a surprising amount of systematic structure lurking within random sequences.  Let’s toss a fair coin where the probability of a head or a tail is exactly .5.

Record the average amount of time (number of tosses) for a pattern to first occur in a sequence (the waiting time). It is significantly long for a repetition (head head — HH or tail tail TT) than for an alternation HT or TH.  The waiting time for HH or TT is 6 tosses, while that for HT or TH is only 4 tosses.  This is in spike of the fact that repetitions and alterations are equally probable (occurring once every four tosses — the same mean time statistics).

This is because repetitions are more bunched in time, they come in bursts, with greater spacing between them, compared with alternations — so they appear less common — hence to the gambler — less probable.

The difference comes from the fact that repetitions can build on each other (e.g. the sequence HHH contains two instances of HH, while alternations cannot).  The waiting time is the variance in the distribution of the interarrival time of patterns which is larger for repetitions than alternations.

So there is a logical basis for the gambler’s fallacy.  Which reminds me.  Did you buy your ticket yet?  Times a’wasting.