The problem doesn't say what it means to crack the bone "at random", but if we assume this means the break is made from a uniform distribution along the length of the bone, then I think the answer is that the average ratio is infinite.
we can assume the bone is of length 1 and we can use symmetry to reduce the problem to a break chosen uniformly on (0,1/2) (it doesn't matter whether the longer piece is on the right or the left)
then if x is that point in the interval (0,1/2) where the break occurs, the ratio of the longer side to the shorter side is: (1-x)/x
so the AVERAGE value of this ratio is... the integral (1-x)/x dx from 0 to 1/2.
but this integral plainly diverges (heck, (1-x)/x will be greater than 0.5/x and the integral of THAT diverges...
So if you believe the uniform distribution is what is meant by cracking the bone "at random", then the answer is that the average ratio of the lengths is infinite, in the sense that as you do the experiment repeatedly, you'd expect the average ratio to grow without bound.