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I've been trying to wrap my head around the oft-quoted misconceptions surrounding the frequentist interpretation of a confidence interval. There are many questions on Cross Validated and many excellent and interesting answers (such as Clarification on interpreting confidence intervals?). It seems to be the general consensus that a 95% confidence interval, for example, should be interpreted in terms of repeating an experiment multiple times and, under such circumstances, the calculated interval will contain the true parameter value 95% of the time. However, it is also clear that most contributors to this site agree that this should not be interpreted as there being a 95% probability that a single confidence interval calculated from a random sample will contain the true (fixed but unknown) parameter value. And the reason appears to be that the frequentist interpretation of a confidence interval relies on long-run frequencies (again see: Clarification on interpreting confidence intervals?). The true parameter value is either in the interval or not and, therefore, probability does not come into it.

Perhaps the reason that I have such a hard time understanding this issue is that it seems to fly in the face of the very earliest statistical concepts that I – and presumably many others – was taught, namely the probability associated with games of chance. If I select a card at random from a well-shuffled standard deck and place it face-down on the table, the probability that the card is a club is 13/52 = 0.25. But, using the same reasoning as that applied to confidence intervals, should I avoid thinking in these terms? The card either is or is not a club and there are no long-run frequencies to consider. So is it legitimate – using frequentist philosophy – to say that the randomly chosen card that I have selected from a deck and placed face down on the table has a 25% probability of being a club?

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There's no need to create a confidence interval for the probability of drawing a club from a deck. That is known. There isn't any uncertainty associated with its value.

You are also confusing probability with observing what has already happened. Before you draw the card has a probability of being a club. (or before you look at a card that is sitting there face down). After you draw the club, you are right it is either a club or is not. That doesn't mean the concept of probability is flawed.

A confidence interval is created for an unknowable quantity. You are estimating something and need to pad your estimate with how close you think you are. You never know the truth. That isn't the case with games of chance where the outcomes are countable and probabilities can be found with 100% certainty.

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    $\begingroup$ The OP is making the argument that the same way we are saying "repeating an experiment multiple times and, under such circumstances, the calculated interval will contain the true parameter value 95% of the time" does not mean "there is a 95% probability that a single confidence interval calculated from a random sample will contain the true value", can be applied to the simple draw a card game. You do not have to explain that a confidence interval is not needed for the pick a card game. You just need to provide an answer why the analogy is problematic. $\endgroup$ – Thanassis Sep 7 '17 at 4:08
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    $\begingroup$ When we draw a card (and we have not seen them yet), we do say that the probability of it being a club is 0.25. But the card is either a club or not. So you need to explain the discrepancy there. The only reason you offer is that in the confidence interval case, the true value is unknowable. What if we could know it through a different process? (imagine we could find the ground truth). Could we talk about probabilities then? $\endgroup$ – Thanassis Sep 7 '17 at 4:13
  • $\begingroup$ "So is it legitimate – using frequentist philosophy – to say that the randomly chosen card that I have selected from a deck and placed face down on the table has a 25% probability of being a club?"...Yes. $\endgroup$ – Michael Sep 7 '17 at 4:28
  • $\begingroup$ If you answer 'yes' to that question you need to explain why the answer is 'no' to the analogous confidence interval question. Your explanation hinges around the unknowable nature of the true parameter value (vs the knowable nature of what the card is). My second comment was questioning this argument. $\endgroup$ – Thanassis Sep 7 '17 at 4:34
  • $\begingroup$ en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat $\endgroup$ – Michael Sep 7 '17 at 4:41
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You know the exact demographics of the population in your card deck scenario. Frequentist statistics, statistics by definition, is used when we don't know the population, but rather we have a sample. That would be like taking X number of cards at random and estimating the probability (proportion) of clubs in the population.

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