I decided to generalize in a different direction, show how you could use a few different programming ideas with this one. Feel free to play around with different sizes, numbers, and choices of die to throw. People who want to play should consider things like using GD::Graph to start producing bar charts etc.

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

my @dist = dice_distribution(
  sides => 6,
  number => 4,
  slice => [1, 2, 3],
);

my $average = sum( map $_*$dist[$_], 0..$#dist ) / sum(@dist);
print "The average is $average\n";

sub dice_distribution {
  my %args = @_;

  my $sides = $args{sides} || 6;
  my $number = $args{number} || 3;
  my $slice = $args{slice} || [0..($number - 1)];

  my @distribution;
  nested_for(
    sub {
      my @dice = (sort {$a <=> $b} @_)[@$slice];
      $distribution[ sum(@dice) ]++;
    }, map [1..$sides], 1..$number,
  );
  # Quiet warnings
  $_ ||= 0 for @distribution;
  return @distribution;
}

sub sum {
  my $sum = 0;
  $sum += $_ for @_;
  return $sum;
}

sub nested_for {
  bind_loops(@_)->();
}

sub bind_loops {
  my $fn = shift;
  my $range = shift;
  my $sub = sub {$fn->($_, @_) for @$range};
  return @_ ? bind_loops($sub, @_) : $sub;
}
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Admittedly I didn't make it shorter. But then again, as pointed out in The path to mastery, I don't think that shorter should always be the immediate aim of someone who is trying to learn...

(Yes, a key part of this bears a suspicious resemblance to Re (tilly) 1 (perl): What Happened...(perils of porting from c). If you are puzzled over how this works, figuring out that first may be a good idea.)