Here is some simpler code that computes the probability that player A tossing a coin $m times gets a greater number of heads than player B tossing a coin $n times, for probability of heads on each toss $p.

#!/usr/bin/perl -w
use strict;

my ($m, $n, $p) = @ARGV;
my $prob = 0;
foreach my $x (1..$m) {
   foreach my $y (0..$x-1) {
      $prob += choose($m, $x) * choose($n, $y) *
         $p**($x + $y) *  (1-$p)**($m + $n - $x - $y);
   }
}
print "$prob\n";

sub choose
{
   my ($n, $k) = @_;
   my $choose = 1;

   foreach my $i (0..$k-1) {
      $choose *= ($n-$i)/($i+1);
   }
   return $choose;
}
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-Mark