Exposure to (western tonal) music means that listeners develop implicit understanding of the rules and regularities in the music and therefore expectations about future events when hearing a new piece :
The idea of implicit knowledge is crucial here (here is a detailed review about implicit musical knowledge). Listeners are sensitive to tonal events and context even though they are not able to label these with the appropriate musical terminology (e.g. dominant-tonic). "Listeners’ tonal knowledge is necessary to interpret differently the same sound event depending on the context in which it occurs. Acculturated listeners understand these musical structures in an implicit way."
Section 2.2 of Tillman's chapter discusses some findings about non-musicians knowledge of tonal relationships.
to understand more about the nature of the evidence you can read more about the methods e.g. probe tone technique.
Note the following observation: "Experimental research on music perception has to find a balance between the complexity of real musical material and strong experimental control of the used material – leading to the use of simple tone contexts or melodies."
Expectation are about what and when.
Evidence of cross-modal influences (section 3.2) suggest that expectations may be guiding our overall attention.
Priming: "The tendency for a stimulus to facilitate the processing of some related ensuing stimulus. For example, when previously presented with the word red a subject will respond faster to most tasks involved in processing the subsequent word blue. That is, the prior presentation of red is said to prime the representation for blue." (Huron)
Section 3.1 in Tillman describes some relevant experimental evidence.
Meyer (and others) suggest that expectations play a significant role in musical expression: "From this play between violations, disruptions and resolutions of expectations raise meaningful and expressive moments in music". But how can expectations be violated when listening to a familiar piece? The suggestion (see section 3.3 for discussion and experimental evidence) is that at least some expectations are an automatic response.
Further reading: review of several works on expectations (including Meyer's work and Narmour's implication-realisation model) from Marcus Pearce.