If the first triangle selected to be shaded is a corner triangle, then the final figure will have at least on axis of symmetry provided that the second triangle selected is one of five triangles. For example, if A is chosen first then there will be at least one axis of symmetry in the final figure if the second triangle selected is B, D, E, G or H. The same applies if an inner triangle is selected first: for example, if B is chosen first then there will be at least one axis of symmetry in the final figure if the second triangle selected is A, C, F, G or H.

So, the probability that the final figure has at least one axis of symmetry is $\frac{5}{7}$.

*This problem is taken from the UKMT Mathematical Challenges.*