  Transition probabilities are an important and useful tool in quantum
mechanics. However, in their present form, they are limited in scope and only
apply to pure quantum states. In this article we extend their applicability to
mixed states and to transitions between quantum effects. We also present their
dependence on a measured operation or instrument. We begin by defining our
concepts on a general quantum effect algebra. These concepts are illustrated
using Holevo operations and instruments. We then present transition
probabilities in the special case of the Hilbert space formulation of quantum
mechanics. We show that for pure states and particular types of operations the
transition probabilities reduce to their usual form. We give examples in terms
of L\"uders operations and instruments.
