Why can't I make this claim about my confidence interval? Say out of a sample of 200 penguins, I find that 192 of them like chocolate. From this sample data, I create the $95\%$ confidence interval for the proportion of penguins that like chocolate:
$$
\left(\frac{192}{200} - 1.96 \times \sqrt{\frac{(\frac{192}{200})(\frac{8}{200})}{200}}, \frac{192}{200} + 1.96 \times \sqrt{\frac{(\frac{192}{200})(\frac{8}{200})}{200}}\right) \\
(0.933, 0.987)
$$
Why would it be invalid for me to subsequently say that I am $95\%$ confident that the percentage of penguins that like chocolate is above $93.3\%$ and below $98.7\%$?
 A: The confidence interval does not rely on any prior probability distribution for the fraction of penguins ($p$) that like chocolate.  You may have thought it extremely unlikely that penguins would like chocolate; if so, you will still deem it unlikely (albeit less so) that the true percentage is between $93.3\%$ and $98.7\%$, even after doing this experiment.  Your confidence that $p$ lies in that interval (that is, your posterior probability) could still be low.
The confidence interval describes something different.  A confidence interval is a rule that assigns an interval to every possible result of your experiment, such that whatever the true value of p, it will fall in the assigned interval $95\%$ of the time.
The difference is fairly subtle, but it can be summarized this way.  A confidence interval is not a particular interval that almost surely (given the randomness associated with $p$) contains the correct answer, now that the experiment has been done.  It is a rule, formulated before the experiment was done, that was almost sure (given the randomness associated with the experiment) to generate an interval containing the correct answer.
