  Since the LHCb collaboration announced the observation of the doubly charmed
baryon $\Xi_{cc}^{++}$, a series of studies of doubly heavy baryons have been
presented. In this work, I analyse the non-leptonic weak decays of doubly heavy
baryons $\Xi_{bc}$ and $\Omega_{bc}$ under the flavor $SU(3)$ symmetry. I
mainly focus on the $W$-exchange diagrams, which will contribute to the decay
channels with final states are light meson and light baryon. These channels
would be helpful for searching for $\Xi_{bc}$ and $\Omega_{bc}$ at LHC. And
these channels and relations of corresponding decay widths could be examined by
the future experimental facilities such as LHC, Belle II and CEPC.
