Here's a problem that could be solved using either analysis or information theory, which approach do you think is easier?
Suppose X_1,X_2,... is an infinite sequence of IID random variables. Let E be an event that's shift invariant, ie, if you take the set of all sequences of Xi's that comprise the event, and shift each sequence, you'll have the same set. For instance "Xi's form a periodic sequence" is a shift-invariant event. Show that P(E) is either 0 or 1.