Consider the following practical algorithm:

sub random_string {
    my $self    = shift;
    my $result  = shift;
    
    $result = ''     unless (defined($result));
    
    my @letters   = ('A'..'Z', 'a'..'z');
    my @letternum = ('A'..'Z', 'a'..'z', '0'..'9');
    
# PRODUCES RANDOM ALPHANUMERIC IDENT 8 CHARS LONG, FIRST CHAR ALPHA.
    $result .= $letters[rand $#letters];
    $result .= join "", map { $letternum[rand $#letternum] } 1..7;
    
    return $result;
}
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One million repetitions later, the same string was never produced twice.   I am quite confident that, if I had ten or even a hundred times as much time to waste waiting for Godot to repeat himself, the result would have been exactly the same.   So, I think that it is quite defensible to say, “it ain’t nevah gonna” happen.   Once you have reduced the probability acceptably close to zero (and of course, have demonstrated in your test-suite that it is, in fact, robust), then ...

“Well, that’s close enough to zero for peace work ...”

Hm. With 52 choices for the first character and 62 thereafter, a 4 character is capable of producing 52 * 62**3 different strings:12,393,056

And yet:

C:\test>junk 4
57147Dup after 57283 reps at C:\test\junk.pl line 18.
57283
C:\test>junk 4
51003Dup after 51478 reps at C:\test\junk.pl line 18.
51478
C:\test>junk 4
2Dup after 23961 reps at C:\test\junk.pl line 18.
23961
C:\test>junk 4
4Dup after 44858 reps at C:\test\junk.pl line 18.
44858
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A tiny part of that is explainable by your using rand( $#array ) instead of rand( @array ) and therefore never picking the final character in those array, giving 51 * 61**3 := 11576031. But the earlyness of the repeat--as low a 24,000--is way too early to be explained even by the Birthday paradox.

Minorly tweaked version of your code used above:

#! perl -slw
use strict;

sub random_string {
    my $n    = shift;
    my $result = '';

    my @letters   = ('A'..'Z', 'a'..'z');
    my @letternum = ('A'..'Z', 'a'..'z', '0'..'9');

    $result .= $letters[ rand $#letters ];
    $result .= join "", map { $letternum[ rand $#letternum ] } 1..$n;

    return $result;
}
my %h;

printf("\r$_"), $h{ random_string( $ARGV[0] ) }++ and die "Dup after $
+_ reps" for 1 .. 1e6;
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---

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.

RIP an inspiration; A true Folk's Guy

a 1 in 30 chance

$ perl -e 'printf "%.3f%", 2**15/1e4'
3.277%
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I know :) I did mention that in the post above.

I also mentioned earlier in the thread, that its use inside a function called truly_random() inside Data::UUID (amongst other things) makes that module suspect.