Accelerate the speed of selection in Random

It is more efficient to use RandomPoint and RandomPoint distributes the points uniformly over the region.

Clear["Global`*"]

f[m_, n_] := {m Cos[n], m Sin[n]};

n = 80000;

To accurately compare the two methods, the point generation needs to be included in the timing.

RepeatedTiming[
 po = RandomVariate[
   UniformDistribution[{{0, 1}, {0, 2 Pi}}], n];
 (Apply[f, po, {1}] // Partition[#, 4] & // 
      Select[#, AllTrue[Last /* GreaterEqualThan[0]]] & // Length)/(n/4) // N]

(* {0.160, 0.06455} *)

Note that the points cluster near the origin.

ListPlot[f @@@ po, AspectRatio -> 1]

RepeatedTiming[
 ((pp = RandomPoint[Disk[], n]) // Partition[#, 4] & // 
      Select[#, AllTrue[Last /* GreaterEqualThan[0]]] & // Length)/(n/4) // N]

(* {0.0423, 0.0619} *)

ListPlot[pp, AspectRatio -> 1]

We get additional speed-up combining RandomPoint with UnitStep,Total and Min or Count:

n = 10^6;

SeedRandom[1];

RepeatedTiming[
 Total[Min /@ UnitStep[RandomPoint[Disk[], {n/4, 4}][[All, All, 2]]]]]

{0.099, 15733}

SeedRandom[1];

RepeatedTiming[
  Count[4] @ Total[UnitStep[RandomPoint[Disk[], {n/4, 4}][[All, All, 2]]], {2}]]

{0.11, 15733}

versus the method from Bob Hanlon's answer:

SeedRandom[1];

RepeatedTiming[
 RandomPoint[Disk[], n] // Partition[#, 4] & // 
   Select[#, AllTrue[Last /* GreaterEqualThan[0]]] & // Length]

{0.8295, 15733}