Two random triangles on six points sharing a corner

Six points in space are in general position (no three on a common line). Alice and Bob each choose, independently and uniformly at random, a set of three of these points to form a triangle; their choices may overlap. What is the probability that the two triangles have at least one vertex in common?

$\tfrac{19}{20}$. The triangles are vertex-disjoint only if Bob picks exactly the three points Alice left out, which is $1$ of the $\binom63=20$ equally likely choices. So the answer is $1-\tfrac1{20}=\tfrac{19}{20}$.