When people are asked to fake randomness, they overdue the randomness
I like this post a lot. People who are asked to fake randomness overdue the “randomness” – they fail to include long sequences where the same result comes over and over again. People naturally understand improbable nature of the coin tosses in Rosencrantz and Guildenstern Are Dead (where one of the characters flips a coin throughout the play, and always gets heads up) but when asked to fake randomness, people go too far in the other direction.
The way to distinguish real random sequences from human-generated ones is to look for a place on the list where there are at least six heads or tails entries in a row. Almost everyone who tries to fake the tosses fails to include a run of such length, yet it is almost a statistical certainty that it will occur in a sufficiently large number of tosses. Using 200 flips, roughly 98% of the entries should have such a sequence of at least six consecutive heads or tails.
This is actually nontrivial to compute (see bottom), but a quick-and-dirty calculation that ignores any conditional probability is as follows: at any given point, the probability of getting six of the same side in a row is (1/2)^6 = 1/64. Thus at any given point the probability of not having such a sequence occur in the next six tosses is (1 – 1/64), and thus the likelihood of this not happening over the entire run is (1 – 1/64)^195 = approximately 5%. (We use 195 not 200, since it is impossible to get six in a row in the last 5 flips). So the rough probability of the sequence happening is 95%. The actual result is even higher.
Hence almost every time we can expect to get a run of six or more, but the near-certainty of this from a probability standpoint does not mesh with our psychological picture of what random coin tosses are supposed to “look like”. In fact, what most people tend to write down is a sort of pseudo-alternation of heads and tails, which is anything but random. If you look at randomness from a compression or signal analysis standpoint, it is equivalent to white noise, meaning that no patterns can be usefully extracted (and no compression can usefully be done). The more a sequence resembles H,T,H,T,H,T…. throughout, the more it becomes nonrandom because it contains the pattern of “H,T alternating”.