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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)
1,400
2,234
3,53

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?

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    $\begingroup$ Google "Shannon mind reading" for one famous solution and various applets that implement it. $\endgroup$ – whuber Jan 6 '17 at 0:18
  • $\begingroup$ After looking up the Mind Reading Machine I am learning a lot of interesting stuff about what Shannon accomplished constructing playing games even as complicated as chess? I am getting the impression that the key to the Mind Reading Machine (which is a binary choice game like coin flipping) is detecting patterns storing them in memory and then looking for them to repeat. This works because humans do not behave randomly. But this coin that changes haphazardly behave like humans? $\endgroup$ – Michael Chernick Jan 6 '17 at 1:07
  • $\begingroup$ No. There may be a pattern or there may not be. The one assumption that I can make is that the trace would be long enough to capture any pattern(s) if one were to exist $\endgroup$ – HXSP1947 Jan 6 '17 at 1:14
  • $\begingroup$ The estimation that I'm looking for only needs to be in the "ball park" so to speak. I'm looking for a heuristic. Specifically I would like to be able to say, given the trace I was provided accurately describes the behavior of the coin, what is the probability that no more than x tails are flipped in a row. This can be accomplished by being able to give an estimation for y or more tails in a row $\endgroup$ – HXSP1947 Jan 6 '17 at 1:16
  • $\begingroup$ I am skeptical about this problem. Even random sequences can look to a human as though there is a pattern. The digits of pi look random but certain prespecified sequence of numbers strings in pi will eventually occur. It has also been said that if you put a monkey at a typewriter hitting keys at random will eventually write a Shakespeare play. $\endgroup$ – Michael Chernick Jan 6 '17 at 1:27

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