According to online sources, if you are operating at 95% confidence, it means if you repeated a sampling process many times and then looked at the 95% confidence intervals over all the results, 95% of the time the brackets would contain the true population mean.
But then they say that it is NOT the same as saying "you can be 95% confident that the intervals you computed contain the population mean."
Isn't it, though? For instance I do my first experiment and get a 95% confidence interval. Then a second. Third, fourth, ..., 100th. 95 of those intervals should contain the population mean. Is this correct so far?
If so, then why isn't it the same as me saying, from the moment I did the very first test, "this particular interval has a 95% chance at being one of the intervals that contain the true population mean"?