Should you believe that this coin is fair?

Faced with a sequence of N binary events, such as coin flips (or Ising spins), it is natural to ask whether these events reflect some underlying dynamic signals or are just random. Plausible models for the dynamics of hidden biases lead to surprisingly high probabilities of misidentifying random sequences as biased. In particular, this probability decays as N^(-1/4), so that no reasonable amount of data would be sufficient to induce the concept of a fair coin with high probability. I suggest that these theoretical results may be relevant to understanding experiments on the apparent misperception of random sequences by human observers.