SetPrecision[Sum[Binomial[n, k] p^k (1. - p)^(n - k) /.
{n -> 50., p -> 10.}, {k,0., 50.}], 1000] // N

With[{pr = 50},
  N[Sum[Binomial[n, k] p^k (1 - p)^(n - k) /. {n -> 50, p -> 10}, {k, 0, 50}], pr]]

With[{pr = 50}, N[Sum[Binomial[50, k] 10^k (1 - 10)^(50 - k), {k, 0, 50}], pr]]