We perform a comprehensive exploration of the Constrained MSSM parameter space employing a Markov Chain Monte Carlo technique and a Bayesian analysis. We compute superpartner masses and other collider observables, as well as a cold dark matter abundance, and compare them with experimental data. We include uncertainties arising from theoretical approximations as well as from residual experimental errors of relevant Standard Model parameters. We delineate probability distributions of the CMSSM parameters, the collider and cosmological observables as well as a dark matter direct detection cross section. The 68% probability intervals of the CMSSM parameters are: 0.52 TeV < m1/2 < 1.26 TeV, m0 < 2.10 TeV, -0.34 TeV < A0 < 2.41 TeV and 38.5 < tan β < 54.6. Generally, large fractions of high probability ranges of the superpartner masses will be probed at the LHC. For example, we find that the probability of mg < 2.7TeV is 78%, of mqR < 2.5TeV is 85% and of mχ±1 < 0.8TeV is 65%. As regards the other observables, for example at 68% probability we find 3.5×10-9 < BR(Bs → μ+μ-) < 1.7×10-8, 1.9×10-10 < δa SUSY μ < 9.9×10-10 and 1 × 10 -10 pb < σSIp < 1 × 10 -8 pb for direct WIMP detection. We highlight a complementarity between LHC and WIMP dark matter searches in exploring the CMSSM parameter space. We further expose a number of correlations among the observables, in particular between BR(Bs → μ+μ-) and BR(B → X sγ) or σSIp. Once SUSY is discovered, this and other correlations may prove helpful in distinguishing the CMSSM from other supersymmetric models. We investigate the robustness of our results in terms of the assumed ranges of CMSSM parameters and the effect of the (g - 2)μ anomaly which shows some tension with the other observables. We find that the results for m0, and the observables which strongly depend on it, are sensitive to our assumptions, while our conclusions for the other variables are robust.
