Remarks on random sequences

We consider evidence relevant to whether a (possibly idealized) physical process is produc-ing its output randomly. For definiteness, we’ll consider a coin-flipper C which reports “H” for heads and “T ” for tails. By C producing its output “randomly, ” we mean H and T have equal probability and trials are independent. If C produces its output randomly (in the above sense), then we’ll say that C is a random device. There are many potential reasons to believe that C is a random device (or that it’s not). We might know something about its manufacture, or be told that C is random on good authority, etc. Our question is whether there is any information in C’s output that bears on whether C is random. We’ll consider two statistics (concerning output sequences generated by C) that are often taken to provide information about C’s randomness. The first statistic is the number of runs in an output sequence [6]. The second statistic is the number of heads versus tails. In order to evaluate the utility of these statistical tests for randomness, we’ll focus on the following two potential output sequences: (A) HTTHTHHH