smc

There are two possible stripe orientations for each of the six faces of the cube, so there are $2^6=64$ possible stripe combinations. There are three pairs of parallel faces so, if there is an encircling stripe, then the pair of faces that does not contribute uniquely determines the stripe orientation for the remaining faces. In addition, the stripe on each face that does not contribute may be oriented in either of two different ways. So a total of $3\cdot 2\cdot 2=12$ stripe combinations on the cube result in a continuous stripe around the cube. The required probability is $\frac{12}{64}=\boxed{\frac{3}{16}}$.