What is the difference between probability and statistics? Is it that probability is top-down (going from pure distributions to predictions about events) and statistics is bottom-up (going from specific events to predicting pure distributions?)
I'm pretty sure that if I have any data involved, then I'm talking about statistics.  But what if I have distributions defined, and want to manipulate them, and real world data is not involved.  Does that mean I'm talking about probability?  
 A: Probability theory is the study of theoretical random processes in which everything is known in principle.  Problems in probability theory involve being given some of the attributes of a theoretical random process and being asked to solve for others that are related.
Statistics is two completely different subjects.
Descriptive statistics is the study of the descriptions and properties of probability distributions themselves, theoretical or empirical.
Inferential statistics is the study of what can be told about a random process of which all or part is unknown, by examining one or more samples.
A: Your first sentence pretty much covers it.  If you toss a coin 100 times and get the same outcome every time, the probabilist can tell you that that's just as likely as any other outcome, whereas the statistician will suspect the coin is biased.  The probabilist can tell you what to infer about the data, given the model; the statistician can tell you what to infer about the model, given the data.
A: I found this resource to be useful:
https://www3.cs.stonybrook.edu/~skiena/jaialai/excerpts/node12.html

This distinction will perhaps become clearer if we trace the thought process of a mathematician encountering her first craps game:

*

*If this mathematician were a probabilist, she would see the dice and think "Six-sided dice? Presumably each face of the dice is equally likely to land face up. Now assuming that each face comes up with probability 1/6, I can figure out what my chances of crapping out are."

*If instead a statistician wandered by, she would see the dice and think "Those dice may look OK, but how do I know that they are not loaded? I'll watch a while, and keep track of how often each number comes up. Then I can decide if my observations are consistent with the assumption of equal-probability faces. Once I'm confident enough that the dice are fair, I'll call a probabilist to tell me how to play."

In summary, probability theory enables us to find the consequences of a given ideal world, while statistical theory enables us to to measure the extent to which our world is ideal.

