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What's the difference between probability and statistics?

Today I just came back from a student seminar in our department. And I heard that for the Pareto distribution, it is kind of difficult to estimate the parameter. There are hundreds of estimations of the parameter and they differ a lot, which means there is no way to estimate it. Thus, I am just wondering what is statistics? Statistics could not estimate such a parameter? Statistics relies what to get the estimation? Not Probability properties of this distribution? Statistics just relies on a mass data set and how could it come up a statistic to estimate the parameter of interest? We use mean to estimate the expectation just because of the central limit theorem, which is totally probability. So what is statistics?

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    $\begingroup$ Related question here $\endgroup$
    – user5594
    Nov 2, 2011 at 23:25
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    $\begingroup$ "There are hundreds of estimations of the parameter and they differ a lot, which means there is no way to estimate it" is rather Zen like: there are so many which mean there are none. $\endgroup$
    – Henry
    Nov 3, 2011 at 0:26
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    $\begingroup$ The body of this message is not entirely coherent because it makes a host of assumptions scattered about with many rhetorical questions that do not seem intended to be answered. Going, then, by the title and the final two lines, it appears that the thrust is what's the difference between probability and statistics, which has already been nicely addressed. If more authoritative information is desired, check out the ASA's answer. $\endgroup$
    – whuber ♦
    Nov 3, 2011 at 4:48

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