Abstract: Hopf algebras are gadgets invented by topologists to study the topology of the classical groups. They are also an emergent 'big deal' in various other areas of mathematics (e.g. combinatorics and analysis). In joint work with Amy Pang and Arun Ram we have discovered that the 'Hopf square' map often has a simple probability interpretation. For example, for the free associative algebra it becomes the usual model for riffle shuffling cards. For the algebra of symmetric functions it becomes a rock breaking model of Kolmogorov. The Hopf machinery now gives an explicit diagonalization leading to sharp rates of convergence. The attempt to deform things to quantum groups leads to interesting open problems. This is an introductory talk with no specialists knowledge assumed.