People often make the mistake of using "crossstream" (inferences from a population of sequencess of a given length, sometimes called "weak") computations to model "downstream" (inferences along a given sequence, sometimes called "strong") computations. Coin tossing (and similar models) are not intuitive at all. For example, in a single game of heads and tails, it is most likely that either H or T is "ahead" most of the time. Crossings are rare.
I knew I was getting in over my head.