  We examine the prospects of detecting the light Higgs h^0 of the Constrained
MSSM at the Tevatron. To this end we explore the CMSSM parameter space with
\mu>0, using a Markov Chain Monte Carlo technique, and apply all relevant
collider and cosmological constraints including their uncertainties, as well as
those of the Standard Model parameters. Taking 50 GeV < m_{1/2}, m_0 < 4 TeV,
|A_0| < 7 TeV and 2 < tan(beta) < 62 as flat priors and using the formalism of
Bayesian statistics we find that the 68% posterior probability region for the
h^0 mass lies between 115.4 GeV and 120.4 GeV. Otherwise, h^0 is very similar
to the Standard Model Higgs boson. Nevertheless, we point out some enhancements
in its couplings to bottom and tau pairs, ranging from a few per cent in most
of the CMSSM parameter space, up to several per cent in the favored region of
tan(beta)\sim 50 and the pseudoscalar Higgs mass of m_A\lsim 1 TeV. We also
find that the other Higgs bosons are typically heavier, although not
necessarily much heavier. For values of the h^0 mass within the 95% probability
range as determined by our analysis, a 95% CL exclusion limit can be set with
about 2/fb of integrated luminosity per experiment, or else with 4/fb (12/fb) a
3 sigma evidence (5 sigma discovery) will be guaranteed. We also emphasize that
the alternative statistical measure of the mean quality-of-fit favors a
somewhat lower Higgs mass range; this implies even more optimistic prospects
for the CMSSM light Higgs search than the more conservative Bayesian approach.
In conclusion, for the above CMSSM parameter ranges, especially m_0, either
some evidence will be found at the Tevatron for the light Higgs boson or, at a
high confidence level, the CMSSM will be ruled out.
