To illustrate their argument they analyse the likelihood of encountering particular sequences of coin tosses. With a sequence of four coin tosses, there are 16 possible sequences that could occur (e.g. HHHH, HHHT, THHT,... etc.). Suppose, though, that we are interested in the occurrences of two equally-likely subsequences, HHH and HHT, within all the possible four-toss outcomes. We can analyse this by representing all 16 sequences in a probability tree diagram, such that HHHH is one branch, HHHT another branch, and so on.

Notice that the subsequence HHH occurs twice within one branch (HHHH), whereas this never happens for HHT. This result generalizes to much longer sequences of coin tosses. Hahn and Warren describe this as being like waiting for a bus: "For a long and frustrating period, there is no bus in sight, and then, all of a sudden, several arrive in immediate succession" (p.455). The upshot of this analysis is that you would need to wait longer (on average) to encounter HHH in a sequence of coin tosses than you would HHT. In a sequence of coin tosses, the expected

*wait time*for HHH is 14 coin tosses, whereas for HHT it is just 8.

The authors also argue that this theoretical analysis maps quite well onto human experience. Limited lifespan and resources mean that random events we encounter are likely to be relatively short sequences, as compared to the

*long runs*discussed in explanations of probability. Furthermore, short-term memory has a very limited capacity, so people can only hold a few events in mind. Hahn and Warren extend their analysis by reporting the results of a computer simulation designed to assess the probability that certain substrings will

*not*occur within a given sequence of coin tosses. They specifically looked at the likelihood of the following substrings: HHHH, HTHH, HHHT, HHTT, and HTHT.

The substring with the highest probability of non-occurrence was HHHH, and the next substring that was likely to not occur was HTHT. Likewise, when Hahn and Warren did the same analysis for substrings of length 6, the most likely substrings to not occur were HHHHHH and HTHTHT. Significantly, previous research has shown that naive participants tend to regard these sequences as less likely to have been generated by a random process than more varied or less regular strings.

In short, Hahn and Warren argue that people's misperceptions of chance are a rational response to the environments that they encounter (although they are still errors). The gambler's fallacy can be viewed in this light also. This fallacy occurs if (say) a gambler believes that a sequence of HHH means that T is more likely to occur on the next toss of the coin. However, Hahn and Warren's analysis shows that a gambler is likely to encounter the sequence HHHT before he or she encounters the sequence HHHH, even though if the gambler has just experienced HHH then the next outcome is equally likely to be H or T.

**Reference**

Hahn, U., and Warren, P.A. (2009). Perceptions of randomness: Why three heads are better than four.

*Psychological Review, 116 (2),*454-461.