The cognoscenti think the average individual is pretty dumb when it comes to probability and randomness. Not so, says a fascinating recent paper [ Proc. Natl. Acad. Sci. vol. 112 pp. 3788 – 3792 ’15 ]. The average joe (this may mean you) when asked to draw a random series of fifty or so heads and tails never puts in enough runs of heads or runs of tails. This leads to the gambler’s fallacy, that if an honest coin gives a run of say 5 heads, the next result is more likely to be tails.
There is a surprising amount of structure lurking within purely random sequences such as the toss of a fair coin where the probability of heads is exactly 50%. Even with a series with 50% heads, the waiting time for two heads (HH) or two tails (TT) to appear is significantly longer than for an alternation (HT or TH). On average 6 tosses will be required for HH or TT to appear while only an average of 4 are needed for HT or TH.
This is why Joe SixPack never puts in enough runs of Hs or Ts.
Why should the wait be longer for HH or TT even when 50% of the time you get a H or T. The mean time for HH and TT is the same as for HT and TH. The variance is different because the occurrences of HH and TT are bunched in time, while the HT and TH are spread evenly.
It gets worse for longer repetitions — they can build on each other. HHH contains two instances of HH, while alterations do not. Repetitions bunch together as noted earlier. We are very good at perceiving waiting times, and this is probably why we think repetitions are less likely and soon to break up.
The paper goes a lot farther constructing a neural model, based on the way our brains integrate information over time when processing sequences of events. It takes into consideration our perceptions of mean time AND waiting times. We average the two. This produces the best fitting bias gain parameter for an existing Bayesian model of randomness.
See, you’re not as dumb as they thought you were.
Another reason for our behavior comes from neuropsychology and physiological psychology. We have ways to watch the electrical activity of your brain and find out when you perceive something as different. It’s called mismatch negativity (see http://en.wikipedia.org/wiki/Mismatch_negativity for more detail). It a brain potential (called P300) peaking .1 -.25 seconds after a deviant tone or syllable.
Play 5 middle c’s in a row followed by a d than c’s again. The potential doesn’t occur after any of the c’s just after the d. This has been applied to the study of infant perception long before they can speak.
It has shown us that asian and western newborn infants both hear ‘r’ and ‘l’ quite well (showing mismatch negativity to a sudden ‘r’ or ‘l’ in a sequence of other sounds). If the asian infant never hears people speaking words with r and l in them for 6 months, it loses mismatch negativity to them (and clinical perception of them). So our brains are literally ‘tuned’ to understand the language we hear.
So we are more likely to notice the T after a run of H’s, or an H after a run of T’s. We are also likely to notice just how long it has been since it last occurred.
This is part of a more general phenomenon — the ability of our brains to pick up and focus on changes in stimuli. Exactly the same phenomenon explains why we see edges of objects so well — at least here we have a solid physiologic explanation — surround inhibition (for details see — http://en.wikipedia.org/wiki/Lateral_inhibition). It happens in the complicated circuitry of the retina, before the brain is involved.
Philosophers should note that this destroys the concept of the pure (e.g. uninterpreted) sensory percept — information is being processed within our eyes before it ever gets to the brain.