  Euclidean volume ratios between quantum states with positive partial
transpose and all quantum states in bipartite systems are investigated. These
ratios allow a quantitative exploration of the typicality of entanglement and
of its detectability by Bell inequalities. For this purpose a new numerical
approach is developed. It is based on the Peres-Horodecki criterion, on a
characterization of the convex set of quantum states by inequalities resulting
from Newton identities and from Descartes' rule of signs, and on a numerical
approach involving the multiphase Monte Carlo method and the hit-and-run
algorithm. This approach confirms not only recent analytical and numerical
results on two-qubit, qubit--qutrit, and qubit--four-level qudit states but
also allows for a numerically reliable numerical treatment of so far unexplored
qutrit--qutrit states. Based on this numerical approach with the help of the
Clauser-Horne-Shimony-Holt inequality and the Collins-Gisin inequality the
degree of detectability of entanglement is investigated for two-qubit quantum
states. It is investigated quantitatively to which extent a combined test of
both Bell inequalities can increase the detectability of entanglement beyond
what is achievable by each of these inequalities separately.
