If you total the weights, pick a rand number between 0 and that total, and then scan the weights, subtracting each from the picked number until it drops below zero; the index of the weight at that point is the ship you want.

sub pick {
    my( $total, @weights ) = @_;
    my $rand = rand $total;
    ( $rand -= $weights[ $_ ] ) < 0
        and return $_ 
        for 0 .. $#weights;
}
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This also works for real weights.

If the number of weights gets to be more than a few tens often, then switching the sub to take an array ref is simple.

And if the weights change frequently, you can do the totalling inside the sub also. I left it external for the test harness.

A little test harness to check it is fair:

#! perl -slw
use strict;
use List::Util qw[ sum ];

our $SHIPS ||= 10;
our $REPS ||= 1e5;

sub pick {
    my( $total, @weights ) = @_;
    my $rand = rand $total;
    ( $rand -= $weights[ $_ ] ) < 0
        and return $_ for 0 .. $#weights;
}

my @weights = map rand( 16 ), 1 .. $SHIPS;
my $total = sum @weights;

my @stats;
for ( 1 .. $REPS ) {
    ++$stats[ pick( $total, @weights ) ];
}

for ( 0 .. $#stats ) {
    printf "%2d: Expected: %6.3f%% Actual: %6.3f%%\n",
        $_,
        $weights[ $_ ] *100 / $total,
        $stats[ $_ ] *100 / $REPS;
}
print sum( @stats ) *100 / $REPS, '%' ;
[download]

Gives:

C:\test>junk6 -SHIPS=4 -REPS=1e5
 0: Expected: 16.519% Actual: 16.604%
 1: Expected: 14.400% Actual: 14.466%
 2: Expected: 51.934% Actual: 51.844%
 3: Expected: 17.147% Actual: 17.086%
100%

C:\test>junk6 -SHIPS=20 -REPS=1e5
 0: Expected:  2.126% Actual:  2.103%
 1: Expected:  6.746% Actual:  6.834%
 2: Expected:  0.847% Actual:  0.825%
 3: Expected:  4.063% Actual:  4.133%
 4: Expected:  4.473% Actual:  4.412%
 5: Expected:  6.020% Actual:  6.079%
 6: Expected:  2.758% Actual:  2.727%
 7: Expected:  5.133% Actual:  5.055%
 8: Expected:  4.221% Actual:  4.215%
 9: Expected:  0.189% Actual:  0.191%
10: Expected:  2.151% Actual:  2.171%
11: Expected:  9.417% Actual:  9.511%
12: Expected:  5.079% Actual:  5.148%
13: Expected:  6.905% Actual:  7.080%
14: Expected:  9.644% Actual:  9.539%
15: Expected:  6.291% Actual:  6.266%
16: Expected:  1.084% Actual:  1.073%
17: Expected:  7.653% Actual:  7.578%
18: Expected:  9.065% Actual:  9.043%
19: Expected:  6.133% Actual:  6.017%

C:\test>junk6 -SHIPS=2 -REPS=1e2
 0: Expected: 92.414% Actual: 96.000%
 1: Expected:  7.586% Actual:  4.000%
100%

C:\test>junk6 -SHIPS=2 -REPS=1e2
 0: Expected: 56.789% Actual: 55.000%
 1: Expected: 43.211% Actual: 45.000%
100%
[download]