One day in high school, my statistics teacher announced an activity: flip a mental coin 25 times (I don’t recall the actual number) and write down the results. Be as random as you can. Then, flip an actual coin 25 times and write down those results too.

Then she walked out of the room. She clearly thought we’d be bad at being random, and so we set about proving her wrong. When she re-entered, every student presented their two sequences of ‘heads’ and ‘tails’:

HTTHHTHHHTHHTTTHTHTH…

HHTHHTTHHHHHTTTHHTTT…

The teacher then strolled around the classroom, stopping at every student’s desk and identifying which sequence was from the true coin and which was from the student’s mind. She did this without fail.

Four years on, this feat still impresses me. How hard is it, really, for humans to be random? And how easily can we tell if randomness is fake? So I sent out this form to some contacts, asking them to submit a sequence of mental coin flips. The theory to test: randomness is more “streaky” than we think. For instance, “HHHHH” is more likely to appear in a truly random sequence of coin flips than it would in a mental-random sequence.

If this theory is true, the probability of switching between Heads and Tails is higher for humans than for random coins. For the 11 responses I received, the red points below mark the probabilities of switching between Heads and Tails for each individual mental flip. The white blobs are null distributions—indications of where these dots tend to be after simulating 10,000 random “coins” via the computer. As you can tell, humans tend to have substantially higher rates of H/T switching than random generators.

Second, the longest runs of continuous Heads or continuous Tails might be shorter in our sequences than in truly random sequences. Blue dots show the longest continuous run for each of the respondents, while the null distributions are again generated by 10,000 simulated random flips:

Hmm, seems substantially lower. We tend to not report any continuous runs longer than “HHHHH” or “TTTTT”, but real random generators tend to spit out runs of up to 7 or 10 continuous Heads. (And the more total flips you do, the longer your longest sequence might be.)

So, next time you’re trying to be random, don’t be afraid of streaks!

Here’s a table of unadjusted one-sided p-values obtained from the simulated null distributions for each respondent:

Respondent Number | # Flips Given | P-Value for Pr(Switch H/T) | P-Value for Longest Run |
---|---|---|---|

1 | 76 | 0.0120 | 0.0700 |

2 | 370 | 0 | 0 |

3 | 424 | 0 | 0 |

4 | 102 | 0.0571 | 0.1846 |

5 | 20 | 0.1816 | 0.2359 |

6 | 65 | 0.0160 | 0.3565 |

7 | 55 | 0 | 0.1566 |

8 | 115 | 0.0061 | 0.1520 |

9 | 62 | 0.7021 | 0.6307 |

10 | 68 | 0.0005 | 0.0895 |

11 | 196 | 0.0134 | 0.9787 |

The data and code for this post are available here. If you’d like to add your own data to the pile, you can submit your mental-random flips here.

To see how streaky random coins are, here are the first 5 random sequences I generated:

HHHTHTTTTHHHTHHHTHTH

TTTHHHTHHHTTHHTTTHHT

TTHHHTTHHHTTHHHTTTTH

THHHHHHTTTTTTTTHTHTT

TTTHTHHTHHHHTHTHTTTH

Awesome post! It was 200 coin flips, by the way, and I always loved the shock and awe I’d get doing this activity. One of my favorite days of teaching!

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