Originally Posted by sureshs
Probability is always used for the future. In the present, it is only relative frequency.
If you want to revise the probabilities based on new data, then you need to apply Bayes theorem and posteriori and a priori probabilities. That is how signal estimation is done in presence of noise as more and more samples arrive.
But the OP didn't mention probability, he mentioned a paradox. A fail (1/n) followed by a success (1/n+1) seems to give greater weight to the second term. Why?
Say a coach has won 9 of 10 games for a 90% win rate. If he wins the next game, his rate becomes 91%, it only goes up 1%. But if he loses, his rate will be 73%, it drops nearly 18%! This doesn't seem fair. With a high success rate, new wins only improve it a tiny bit, but just one loss makes a huge drop. This lack of symmetry between wins and losses explains the puzzle the OP discussed, IMO.