In addition to H7, Hy and AC, if computable in a meaningful way, the

In addition to H7, Hy and AC, if computable in a meaningful way, the deviation of the binary series under study from the idealized random behavior of a fair coin could serve as another index of complexity. Common descriptions of the amount of randomness in a series are indices of run length If a run length is defined by number of elements in a series of the same symbol before it stops, counting the number of run boundaries by reading along the binary series and counting the number of switches from 0 —+1 or 1 — 0, then the binary expression of 729, 1011011001, has six runs. The great analytic probabilist, William Feller, among many others, including the distinguished 18 Century Swiss family of mathematicians, the Bernoulli's, proved that computing a standard run score, srs, involves three terms, the theoretical expectation, E, of the number of runs, r, that is E(r), the number of runs actually observed, Obs(r) and the variance of the expectation of the number of runs, Var( E(r). If the srs is less than zero, then the binary series is more random than that resulting from the flipping of a fair coin. Interestingly, when a normal group of subjects are instructed to simulate what they think of as a random coin flip determined series of 0’s and 1’s, their srs tends to be lower than zero, over-estimating the degree of irregularity that randomness represents. Long runs occur by chance far more often than intuition would dictate. If srs is more than zero, than the binary, coin-flip series is more ordered than random. If srs equal to zero, the binary series is not discriminable from fair coin flipping randomness. The expected number of runs, E(r), can be estimated by a fraction formed by twice the product of the number of heads times tails divided by the sum of the 2x6x4 heads and tails to which is added one. That is, E(r) = ZA + +1=5.8. The average variation around this expectation called the variance, Var, of the expectation, Var(E(r)), is estimated by a fraction formed by (take a breath) twice the product of the number of heads times tails x twice the product of the number of heads times tails minus the number of heads and minus the number of tails, all over the product of the sum of the heads and tails squared, times the sum of the number of heads 111 HOUSE_OVERSIGHT_013611