Rather than brute force iteration, you could always simulate:

#! perl -slw
use strict;
use Data::Dump qw[ pp ];
use List::Util qw[ sum ];
use Math::Random::MT qw[ rand ];

my $throws   = 5;
my $sides    = 5;

my @probs = ( 0.05, 0.1, 0.2, 0.3, 0.35 );
my @values = ( 1, 5, 7, 13, 17 );
my @lookup = map{
    ( $values[ $_ ] ) x ( $probs[ $_ ] * 100 );
} 0 .. $#values;

our $MIN  //= 60;
our $RUNS //= 1e5;

my $moreThanMin = 0;
for ( 1 .. $RUNS ) {
    my $total = sum map{ $lookup[ rand( scalar @lookup ) ] } 1 .. $thr
+ows;
    ++$moreThanMin if $total > $MIN;
}
printf "Probability based on $RUNS simulations: %.3f%%",
    $moreThanMin / $RUNS * 100;

__END__

c:\test>895892 -RUNS=1e6
Probability based on 1e6 simulations: 48.752%
[download]

It won't save time on small die, but on the larger ones you can decide how much time you want to spend to get closer and closer to the answer.