  Incompatibility of quantum measurements is of fundamental importance in
quantum mechanics. It is closely related to many nonclassical phenomena such as
Bell nonlocality, quantum uncertainty relations, and quantum steering. We study
the necessary and sufficient conditions of quantum compatibility for a given
collection of $n$ measurements in $d$-dimensional space. From the compatibility
criterion for two-qubit measurements, we compute the incompatibility
probability of a pair of independent random measurements. For a pair of
unbiased random qubit measurements, we derive that the incompatibility
probability is exactly $\frac35$. Detailed results are also presented in
figures for pairs of general qubit measurements.
