The mathematicians who analyzed coin-flipping sequences for patterns were not expecting to find them. The coin flip is the standard example of a random event — the archetype of a process in which past outcomes provide zero information about future outcomes. The probability of heads on any given flip is 0.5, regardless of whether the previous ten flips were all tails.

This is true. And yet researchers studying sequences of coin flips produced by people versus sequences produced by actual coins found that people are dramatically worse at generating genuinely random sequences than they believe. When asked to produce a sequence that "looks random," people systematically under-produce long runs of the same result — because runs of five or six heads feel non-random to the human pattern-recognition system — and over-produce alternations. Genuine random sequences contain more long runs than human intuition expects.

This finding has been applied in various contexts, including the detection of fabricated data in research — graduate students who are tempted to adjust their experimental results toward "cleaner" outcomes tend to produce data with fewer extreme values and more regular patterns than genuinely random experimental processes would generate.

The coin-flip game that can be won exploits a specific mathematical asymmetry discovered by Walter Penney and subsequently analyzed extensively. If two players each choose a sequence of heads and tails — say, one chooses HHT and the other chooses THH — and then flip coins until one sequence appears, the player who knows this asymmetry can almost always choose the sequence that wins more than half the time. The game is unfair in a non-obvious way that the mathematics reveals and intuition misses entirely.

The implications for gambling, for finance, and for any domain where people are making predictions about apparently random sequences are left as an exercise for the reader.