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I have read two books that explicitly state that the $(1-\alpha)$% confidence interval should be interpreted as:

If you construct 100 such confidence intervals, $\alpha$ of them are expected to not contain the true population statistic and $(1-\alpha)$ of them are expected to contain the true population statistic.

and not as

There is a $(1-\alpha)$% probability that the true population statistic is contained in the confidence interval.

In my view, they both equivalent: If you make the first statement, you implicitly make the second statement. You are looking at any one arbitrary confidence interval, which in itself is a random variable, the generic confidence interval should, therefore, be subject to the second statement. Do things change when this random variable is actually realized?

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I struggled with this concept for quite a while, but an intuitive explanation for it I found on Wikipedia (as background context a $95\%$ confidence interval of the number of voters voting for a particular party was found to be $[36\%,44\%])$:

For example, in the poll example outlined in the introduction, to be 95% confident that the actual number of voters intending to vote for the party in question is between 36% to 44%, should not be interpreted in the common-sense interpretation that there is a 95% probability that the actual number of voters intending to vote for the party in question is between 36% to 44%. This would be technically incorrect. The actual meaning of confidence levels and confidence intervals is rather more subtle. In the above case, a correct interpretation would be as follows: If the polling were repeated a large number of times (you could produce a 95% confidence interval for your polling confidence interval), each time generating about a 95% confidence interval from the poll sample, then approximately 95% of the generated intervals would contain the true percentage of voters who intend to vote for the given party. Each time the polling is repeated, a different confidence interval is produced; hence, it is not possible to make absolute statements about probabilities for any one given interval.

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I think confidence intervals are better understood in the following way. There is an infinite set of constant open intervals. Some of them cover the population statistic $\theta$ and some of them don't. Now the (random) confidence interval is just a mechanism to draw an interval from this set. So the $(1-\alpha)\%$ is the chance of picking a constant interval that covers $\theta$. But whatever interval is drawn (or realized), since each of these interval is constant, the probability that it contains $\theta$ can only be $0$ or $1$.

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