  We present a rigorous method, based on Bayesian inference, for calculating
the odds favoring the hypothesis that any particular class of astronomical
transients produce gamma-ray bursts over the hypothesis that they do not. We
then apply this method to a sample of 83 Type Ia supernovae and a sample of 20
Type Ib-Ic supernovae. We find overwhelming odds against the hypothesis that
all Type Ia supernovae produce gamma-ray bursts, whether at low redshift
($10^{9}:1$) or high-redshift ($10^{12}:1$), and very large odds ($6000:1$)
against the hypothesis that all Type Ib, Ib/c, and Ic supernovae produce
observable gamma-ray bursts. We find large odds ($34:1$) against the hypothesis
that a fraction of Type Ia supernovae produce observable gamma-ray bursts, and
moderate odds ($6:1$) against the hypothesis that a fraction of Type Ib-Ic
supernovae produce observable bursts. We have also re-analyzed both a corrected
version of the Wang & Wheeler sample of Type Ib-Ic SNe and our larger sample of
20 Type Ib-Ic SNe, using a generalization of their frequentist method. We find
no significant evidence in either case of a correlation between Type Ib-Ic SNe
and GRBs, consistent with the very strong evidence against such a correlation
that we find from our Bayesian analysis.
