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But now say we have knowledge of the first set of flips and we see that 7 out of 10 of the flips were heads (but we are still convinced we have a fair coin). Now the next person who comes in expects to see 5 heads on their set of 10 flips and expects to see 50 heads overall as most likely because they are unaware of the first set and are starting from a baseline of 0. But we have the knowledge of what happened in the past. And we now do NOT expect that 50 heads is the most likely outcome. Because the most likely outcome is 7 heads (that we've observed) plus the most likely outcome over the next 9 sets or 90 coin flips. That is 7 heads plus 45 heads. So we, because we know what just happened and aren't starting at a baseline of 0, know that at that moment in time the most likely number of heads to end up with is 52.


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"And we now do NOT expect that 50 heads is the most likely outcome." This statement is wrong: before the sequence you observed there could have been a series of 500 (or whatever) coins flips, all tails.

If this is not an issue, then you accept implicitely that one sequence of events (the first one) can be biased, but not any other (inlcuding a longer one)???

Your arguments only holds with big numbers and they are numbers that get close to infinity, not the numbers you are using.

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It seems like you completely missed the entire logic of SumZero's post. If you know a coin will be flipped 100 times, you expect to get 50 heads. If the first ten flips result in 7 heads, you now expect to get 45 heads on the remaining 90 flips, plus the seven you already have, for a total of 52 heads.

What happens before the observed sequence does not matter at all. All that matters is that we have a fair coin and a set number of trials. We can then adjust our estimation of the final number of heads based on the total number of heads so far, which would look like: (number of heads) + ((50%) * (remaining number of flips)).
As you should be able to see, this fits exactly with what is in SumZero's post.