  In the operator formalism of quantum mechanics, the density operator
describes the complete statistics of a quantum state in terms of d^2
independent elements, where d is the number of possible outcomes for a precise
measurement of an observable. In principle, it is therefore possible to express
the density operator by a joint probability of two observables that cannot
actually be measured jointly because they do not have any common eigenstates.
However, such joint probabilities do not refer to an actual measurement
outcome, so their definition cannot be based on a set of possible events. Here,
I consider the criteria that could specify a unique mathematical form of joint
probabilities in the quantum formalism. It is shown that a reasonable set of
conditions results in the definition of joint probabilities by ordered products
of the corresponding projection operators. It is pointed out that this joint
probability corresponds to the quasi probabilities that have recently been
observed experimentally in weak measurements.
