Completely understanding the Confidence Interval - how is it not a probability of containing a population parameter?

I've been doing data analysis for a while, but recently I questioned my understanding of the oft-misunderstood confidence interval. So, I read multiple sources.

Many of them say explicitly that the confidence interval is NOT the probability that a specific confidence interval contains the population parameter. Rather, it is the percentage of times a confidence interval will contain the population parameter after many samples.

I honestly can't tell the difference between these: Isn't the latter the same as the probability? If not, how is it different? I suppose the definition of probability might be throwing me off here.

Here's one of the sources I used: http://blog.minitab.com/blog/adventures-in-statistics/understanding-hypothesis-tests%3A-confidence-intervals-and-confidence-levels

• Please try reading this Cross Validated page, which is directly on this topic, and edit your question if you need further clarification on specific issues. – EdM Dec 27 '15 at 16:29
• see section 3 of this answer: stats.stackexchange.com/questions/167972/… – user83346 Dec 27 '15 at 18:39
• @Candic3 In response to your paragraph "I honestly can't tell the difference between these: Isn't the latter the same as the probability? If not, how is it different?": Both statements are indeed probabilities, but they are probabilities of different things. In the correct definition, the probability refers to the confidence intervals; in the incorrect definition, the probability refers to the parameter. (I've got more detail in the Day 2 slides at ma.utexas.edu/users/mks/CommonMistakes2015/…) – Martha Dec 27 '15 at 21:20