Decays of $B^0$ and $B_{s}^0$ mesons to the charmless baryonic final states $p overline{p} h^+ h^{prime-}$, where $h$ and $h^prime$ each denote a kaon or a pion, are searched for using the LHCb detector. The analysis is based on a sample of proton-proton collision data collected at center-of-mass energies of $7$ and $8,$TeV, corresponding to an integrated luminosity of $3,$fb$^{-1}$. Four-body charmless baryonic $B_{s}^0$ decays are observed for the first time. The decays $B^0_{s}to p overline{p} K^+ K^-$, $B^0_{s}to p overline{p} K^pm pi^mp$, $B^0to p overline{p} K^pm pi^mp$ and $B^0to p overline{p} pi^+pi^-$ are observed with a significance greater than $5$ standard deviations; evidence at $4.1$ standard deviations is found for the $B^0to p overline{p} K^+ K^-$ decay and an upper limit is set on the branching fraction for $B^0_{s}to p overline{p} pi^+ pi^-$. Branching fractions in the kinematic region $m(p overline{p})<2850,$MeV$/c^2$ are measured relative to the $B^0 to J/psi (to p overline{p}) K^*(892)^0$ channel.
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