Interesting probability question

[Siegmund]

Here's a somewhat simpler way of making the same calculation:

Look at each of the elements 1,2,3...n in turn.

Four different things, equally likely, can happen to Element 1: it can be in neither set, it can be in A but not B, it can be in B but not A, or it can be in both.
One of these four things prevents A from being a subset of B, the other three do not.