I’ve read through this thread, and it looks to me like it can be said that:
 statistics = induction ?
 probability = deduction ?
But I am wondering if there might be some more details on the comparison that I am missing. For example, is statistics equal to induction, or is it just a particular case of it? It does seem that probability is a sub case of deduction (since it is a sub case of mathematical thinking).
I know this is a picky question, but in a sense this is why I am asking it – because I want to be sure how these terms can be compared accurately.
Answer
I think it is the best to quickly recap the meaning of inductive and deductive reasoning before answering your question.

Deductive Reasoning: “Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises. A deductive argument is valid if the conclusion does follow necessarily from the premises, i.e., if the conclusion must be true provided that the premises are true. A deductive argument is sound if it is valid and its premises are true. Deductive arguments are valid or invalid, sound or unsound, but are never false or true.” (quoted from wikipedia, emphasis added).

“Inductive reasoning, also known as induction or inductive logic, or educated guess in colloquial English, is a kind of reasoning that allows for the possibility that the conclusion is false even where all of the premises are true. The premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they do not ensure its truth.” (from wikipedia, emphasis added)
To stress the main difference: Whereas deductive reasoning transfers the truth from premises to conclusions, inductive reasoning does not. That is, whereas for deductive reasoning you never broaden your knowledge (i.e., everything is in the premises, but sometimes hidden and needs to be demonstrated via proofs), inductive reasoning allows you to broaden your knowledge (i.e., you may gain new insights that are not already contained in the premises, however, for the cost of not knowing their truth).
How does this relate to probability and statistics?
In my eyes, probability is necessarily deductive. It is a branch of math. So based on some axioms or ideas (supposedly true ones) it deduces theories.
However, statistics is not necessarily inductive. Only if you try to use it for generating knowledge about unobserved entities (i.e., pursuing inferential statistics, see also onestop’s answer). However, if you use statistics to describe the sample (i.e., decriptive statistics) or if you sampled the whole population, it is still deductive as you do not get any more knowledge or information as that is already present in the sample.
So, if you think about statistics as being the heroic endeavor of scientists trying to use mathematical methods to find regularities that govern the interplay of the empirical entities in the world, which is in fact never successful (i.e., we will never really know if any of our theories is true), then, yeah, this is induction. It’s also the Scientific Method as articulated by Francis Bacon, upon which modern empirical science is founded. The method leads to inductive conclusions which are at best highly probable, though not certain. This in turn leads to misunderstanding among nonscientists about the meaning of a scientific theory and a scientific proof.
Update: After reading Conjugate Prior’s answer (and after some overnight thinking) I would like to add something. I think the question on whether or not (inferential) statistical reasoning is deductive or inductive depends on what exactly it is that you are interested in, i.e., what kind of conclusion you are striving for.
If you are interested in probabilistic conclusions, then statistical reasoning is deductive. This means, if you want to know if e.g., in 95 out of 100 cases the population value is within a certain interval (i.e., confidence interval) , then you can get a truth value (true or not true) for this statement. You can say (if the assumptions are true) that it is the case that in 95 out of 100 cases the population value is within the interval. However, in no empirical case you will know if the population value is in your obtained CI. Either it is or not, but there is no way to be sure. The same reasoning applies for probabilities in classical pvalue and Bayesian statistics. You can be sure about probabilities.
However, if you are interested in conclusions about empirical entities (e.g., where is the population value) you can only argue inductive. You can use all available statistical methods to accumulate evidence that support certain propositions about empirical entities or the causal mechanisms with which they interact. But you will never be certain on any of these propositions.
To recap: The point I want to make that it is important at what you are looking. Probabilites you can deduce, but for every definite proposition about things you can only find evidence in favor for. Not more. See also onestop’s link to the induction problem.
Attribution
Source : Link , Question Author : Tal Galili , Answer Author : Fredric100