eh i went ahead and did this brilliant.org/community-prob
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In general it is (total number of tails so far +1)/(total number of flips so far +2)
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You mean if it is from range [p,q] not [0,1]? Then integrals become messy and there is no neat and short formula I think.
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When it's not uniformly taken from [0,1], with some number more likely than others.

