Given a random sample $x$ and wishing to estimate the value of mean $\mu$ of a population from which $x$ was drawn, I construct a 95% confidence interval $CI_{x}$ using $\mu_{x}$ and $\sigma_{x}$. If I repeat this procedure an infinite number of times, drawing different samples, 95% of these intervals will contain the true population mean $\mu$.
I fail to understand why it is incorrect to say that the interval constructed using one sample has a 95% chance (or probability) of containing $\mu$.
As an analogy, in a generous lottery 95% of tickets are winners, and I just bought a ticket, my ticket has a 95% chance of being a winning ticket.
I've seen arguments that once a sample is drawn and a confidence interval calculated, the interval does or does not contain the true mean. There is no associated probability. Going back to the lottery, one could make a similar argument. But as I don't yet know the results of the lottery, why can't I say that my ticket has a 95% chance of being a winning ticket?
