  The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow
\{0,1\}$ is defined to be the smallest $d$ such that there is a random
polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each
point with high probability. Introduced by Razborov (1987), upper and lower
bounds on probabilistic degrees of Boolean functions --- specifically symmetric
Boolean functions --- have been used to prove explicit lower bounds, design
pseudorandom generators, and devise algorithms for combinatorial problems.
  In this paper, we characterize the probabilistic degrees of all symmetric
Boolean functions up to polylogarithmic factors over all fields of fixed
characteristic (positive or zero).
