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Beginner question here. I've read (see this paper) that when computing confidence intervals if you get a given interval with a confidence level of 95%, then it is a fallacy to think that the true parameter is inside that confidence interval with 95% of confidence. The paper linked above makes a case and explains this, but I'm afraid I don't get the intution.
For instance, let's say that I randomly sampled from a population 1,000,000 times, and that I'm sure that 95% of the resulting confidence intervals will contain the population parameter. So I know that 950,000 confidence intervals contain the population parameter, and that 50,000 don't.
It seems fairly reasonable to me that if I pick up one of those confidence intervals, there is a 95% chance that I will have one of the intervals that do contain the true population parameter. So, if I sampled just once from the population and calculated a confidence interval, then it will count as a random sample from the set of possible intervals. I am 95% sure that this interval contains the true population parameter.
Kristoffer Magnusson has a nice web app (here) where he seems to suggest that there is indeed a fallacy in my kind of reasoning. So several people say that this is a basic thinking flaw, but I don't really get it. Any help will be most welcome.