that's NOT true ... but That was a joke right ? Beware you might plant some weird idea on ppl mind, probas are tricky enough as they are

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In theory, there is no difference between theory and practice. In practice, there is.

@ percival
I think we both read the question wrong. I (and probably you too) interpreted the question as; Flip a coin 3 times, which sequence is more likely: HHT or HTH. They are obviously the same with each being 1/8. This is because each coin flip is a separate event and the result is not influenced by previous flips. You can easily draw a decision tree and then you just have to walk along one of the branches and multiply the probabilities.

But the question was: I play a large number of games (10000+) which sequence is likely to appear first and how often will it appear. This kind of math is more advanced than the first one because it depends on the previous results. If the first two flips are HH then you have 50% to hit HHT but 0% to hit HTH

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While I was teaching the game to a friend of mine, my mother from the other room:
"Cutting? Killing? Poking out eyes? What the hell are you playing?"

i agree with that but my "that's NOT true" was to jts first comment: saying that if you got 1 H first you have less than 1/2 prob to have H next to "maintain the balance" but i think that was a joke . Or i didnt understand the statement
Hyperpape explanation is the right one (IMHO) even if it still hurts my common sense so much that i want to find a flaw.

i'll come up with my own Proba anecdote:

There was a probability teacher who would ask his class, onthe first day, to generate a random string of H and T in two ways:
Th
The one whose mother last name started with an L or before should flip a coin 200 times, the other should just make out a sequence from the top of their heads.
the "mother last name" thing is choosen so that the professor doesnt know which student used which way (but still some students will have to).

Then the professor would guess which one was simulated and which wasnt at a glance wit ha very high sucess rate. How did he do that ?

.. i don t remember where i heard that story hope it wasnt on that board....

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In theory, there is no difference between theory and practice. In practice, there is.

Back to Araban Question, here is a more complete version of hyperpape answer; (with the inital 3 valued event)

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In theory, there is no difference between theory and practice. In practice, there is.