Thanks in advance for the help.
Suppose someone has an unbalanced coin that they flip 100,000 or so times in a row. This person then gives you the results. You do not know the probability of getting a heads or tails by flipping the coin the person used; all you have is the trace of results from 100,000 flips. In addition, the coin that they used is not independent. That is, every single coin flip that proceeded any arbitrary flip may an effect on the outcome of the arbitrary flip (some may not effect the outcome, you don't know. A markov chain may be the correct way to think about this, I'm not sure). You, however, are not aware of any of these effects besides knowing they exist. All you have is the trace.
Now suppose that you get access to this coin and you would like to predict what would happen should you flip it, given only your knowledge of the trace. In particular, suppose you want to flip the coin until you get a head. I'm interested in the probability of getting one or more tails, two or more tails, three or more tails, etc.
Now, I obviously cannot determine the exact probabilities since I don't know the probabilities of the unbalanced coin nor the how previous flips effect the current flip. I should, however, be able to give an estimation (though I don't believe I will know how accurate that estimation is). What would be a reasonable way of doing this?
This is what I've been thinking. I can first count how many times I get one tail, two tails, three tails, etc in a row within the trace (lets say tails occur much less frequently than heads). ie
tails in a row, number of occurrences (this implies 98973 heads and 1027 tails)
The probability of getting one or more tails in a row then might look like
(98973 / 100000) * ((400 + 234 + 53) / 100000)
since getting x tails in a row roughly implies a head was flipped before the tail. So the probability would be the possibility of the start of a series of tails being flipped multiplied by the possibility of getting a head. Does this makes sense and is it remotely reasonable? If not, what might be a better a better approach to get an estimation of what I'm looking for?