Recently, Mike and I have found ourselves discussing several research studies with supposedly “surprising” results. While the work involved appears solid and the details of the phenomena were certainly amazing, we struggled to understand why the general phenomenon was “surprising”. Amazing, yes. Unexpected, no. After years of such discussions, Mike has asked me to write down Witten’s Rules of Astonishment*.
WITTEN’S RULES OF ASTONISHMENT
I. Although unlikely events are statistically improbable, there are a shitload of independent events in any given unit of time.
a. People systematically overestimate the improbability of unlikely events and underestimate the number of events.
II. Surprise is usually a failure of imagination.
III. “Amazed” is not the same as “surprised”.
*Though independently derived, I make no claim that these “rules” do not overlap with the “rules” of other individuals, particularly those that may have established theirs first.
The Finch and Pea 
So would you say that “surprise” is properly limited to situations where we had reason to believe that something was untrue/impossible? And “amazement” is when something was unexpected, but is easy to believe once it has been announced?
I’d say “surprise” is for when something does not fit expectations about how the world works. “Amazed” is for being impressed by the variety of things that are possible within the way the world works.