I've got a problem where I need to probabilistically score different possible sequences of events, and I think I'm missing something about how to properly model them. Here's the example: Say someone flips a fair coin once per second for a whole day. I didn't seen the coin flips, but am given a two possible sequences. One has ~50% heads, but they all occur up front (the first half of the flips are all heads, then the remainder are all tails). The second sequence is also ~50% heads, but they occur in a more random seeming way, distributed throughout the day. In what sense, if any, is the second sequence more likely?
At the sequence level, all sequences have equal likelihood. I can also look at a binomial distribution in number of heads, but they would score equally well against that metric too. I could also look at the event arrival times, in terms of the number of flips between heads, perhaps looking at it as a Poisson with an exponential distribution describing the time between changes. I believe this ends up scoring the same too though. An ad-hoc thing would be to break it into time segments and score them separately - that would expose that it's very unlikely to get zero heads in 12 hours of flipping, but I don't like the ad-hoc nature of it, where I need to pick a time window interval.
What's the right way to do this?