BrowserUk:

Heh, yep, that would be a good deal slower, all right! Of course, if speed gets to be an issue, you can always skip counting all the iterations, and directly compute the binomial expansion:

use strict;
use warnings;
use List::Util qw(reduce);

my $numTosses = 20;
#my $runs = (1<<$numTosses)-1;

my @triangle = (0, 1, 0);
for (1 .. $numTosses) {
    my @newTriangle=(0);
    push @newTriangle, $triangle[$_]+$triangle[$_+1] for 0 .. $#triang
+le-1;
    push @newTriangle, 0;
    @triangle = @newTriangle;
}

print <<EOHDR;
Tails      Count     %
-----   ----------  ------
EOHDR
my $runs = reduce { $a + $b } @triangle;
for (my $i = 0; $i < $numTosses+1; $i++) {
    printf "% 4u    % 10u   %5.2f\n",
            $i, $triangle[$i+1], 100*$triangle[$i+1]/$runs;
}
[download]

It runs a good bit faster:

$ time perl 892293.pl
Name "main::a" used only once: possible typo at 892293.pl line 21.
Name "main::b" used only once: possible typo at 892293.pl line 21.
Tails      Count     %
-----   ----------  ------
   0             1    0.00
   1            20    0.00
   2           190    0.02
   3          1140    0.11
   4          4845    0.46
   5         15504    1.48
   6         38760    3.70
   7         77520    7.39
   8        125970   12.01
   9        167960   16.02
  10        184756   17.62
  11        167960   16.02
  12        125970   12.01
  13         77520    7.39
  14         38760    3.70
  15         15504    1.48
  16          4845    0.46
  17          1140    0.11
  18           190    0.02
  19            20    0.00
  20             1    0.00

real    0m0.034s
user    0m0.024s
sys     0m0.012s
[download]

Even if you use 32 bits:

$ time perl 892293.pl
Name "main::a" used only once: possible typo at 892293.pl line 21.
Name "main::b" used only once: possible typo at 892293.pl line 21.
Tails      Count     %
-----   ----------  ------
   0             1    0.00
   1            32    0.00
   2           496    0.00
   3          4960    0.00
   4         35960    0.00
   5        201376    0.00
   6        906192    0.02
   7       3365856    0.08
   8      10518300    0.24
   9      28048800    0.65
  10      64512240    1.50
  11     129024480    3.00
  12     225792840    5.26
  13     347373600    8.09
  14     471435600   10.98
  15     565722720   13.17
  16     601080390   13.99
  17     565722720   13.17
  18     471435600   10.98
  19     347373600    8.09
  20     225792840    5.26
  21     129024480    3.00
  22      64512240    1.50
  23      28048800    0.65
  24      10518300    0.24
  25       3365856    0.08
  26        906192    0.02
  27        201376    0.00
  28         35960    0.00
  29          4960    0.00
  30           496    0.00
  31            32    0.00
  32             1    0.00

real    0m0.034s
user    0m0.028s
sys     0m0.004s
[download]

...roboticus

When your only tool is a hammer, all problems look like your thumb.

Update: When I use bignum;, it prints the PDF for 64 tosses per run in just over a second, but I haven't got it formatted nicely...

use strict;
use warnings;
use List::Util qw(reduce);
use bignum;

my $numTosses = 64;
#my $runs = (1<<$numTosses)-1;

my @triangle = (0, 1, 0);
for (1 .. $numTosses) {
    my @newTriangle=(0);
    push @newTriangle, $triangle[$_]+$triangle[$_+1] for 0 .. $#triang
+le-1;
    push @newTriangle, 0;
    @triangle = @newTriangle;
}


print <<EOHDR;
Tails      Count     %
-----   ----------  ------
EOHDR
my $runs = reduce { $a + $b } @triangle;
for (my $i = 0; $i < $numTosses+1; $i++) {
    print "$i\t$triangle[$i+1]\t",100*$triangle[$i+1]/$runs, "\n";
}
[download]

I guess I'll have to figure out how to make bugnum and printf cooperate better...