  Quantum theory does not provide a unique definition for the joint probability
of two non-commuting observables, which is the next important question after
the Born's probability for a single observable. Instead, various definitions
were suggested, e.g. via quasi-probabilities or via hidden-variable theories.
After reviewing open issues of the joint probability, we relate it to quantum
imprecise probabilities, which are non-contextual and are consistent with all
constraints expected from a quantum probability. We study two non-commuting
observables in a two-dimensional Hilbert space and show that there is no
precise joint probability that applies for any quantum state and is consistent
with imprecise probabilities. This contrasts to theorems by Bell and
Kochen-Specker that exclude joint probabilities for more than two non-commuting
observables, in Hilbert space with dimension larger than two. If measurement
contexts are included into the definition, joint probabilities are not anymore
excluded, but they are still constrained by imprecise probabilities.
