Computing the Effect of Advantage
There are mechanisms in D&D 5e, Advantage and Disadvantage. Basically, Advantage means, when you roll, you roll twice and keep the better roll. Disadvantage means the opposite; you roll twice and keep the worst.
Last I played was AD&D in the 80s, so new could still be decades old, but I’ve recently started watching Dimension 20 and becoming more familiar with what happened after I became an adult with responsibilities.
So, after understanding why it would come up, I decided to understand the result. I wrote code that made the rolls 10 million times, then showed me the results.
Normal
NORMAL
ROLL CNT MAX AVG PERC BLOCK
1 500336 501640 10 5.0034 #####
2 498744 501640 10 4.9874 ####
3 499964 501640 10 4.9996 ####
4 499328 501640 10 4.9933 ####
5 500378 501640 10 5.0038 #####
6 498710 501640 10 4.9871 ####
7 499798 501640 10 4.9980 ####
8 499039 501640 10 4.9904 ####
9 499554 501640 10 4.9955 ####
10 499196 501640 10 4.9920 ####
11 501384 501640 10 5.0138 #####
12 499154 501640 10 4.9915 ####
13 501640 501640 10 5.0164 #####
14 500112 501640 10 5.0011 #####
15 501159 501640 10 5.0116 #####
16 500273 501640 10 5.0027 #####
17 500893 501640 10 5.0089 #####
18 499613 501640 10 4.9961 ####
19 499946 501640 10 4.9995 ####
20 500779 501640 10 5.0078 #####
This is random, so there’s variable, but every number is within a pebble toss from being a 5% chance, and seeing that 100/20 == 5, that tracks, as does the average being right at the center. So, the analysis works.
Advantage
ADVANTAGE
ROLL CNT MAX AVG PERC BLOCK
1 24970 975534 13 0.2497
2 75003 975534 13 0.7500
3 125124 975534 13 1.2512 #
4 174964 975534 13 1.7496 #
5 225133 975534 13 2.2513 ##
6 275264 975534 13 2.7526 ##
7 323521 975534 13 3.2352 ###
8 375226 975534 13 3.7523 ###
9 425252 975534 13 4.2525 ####
10 474704 975534 13 4.7470 ####
11 526639 975534 13 5.2664 #####
12 575093 975534 13 5.7509 #####
13 623744 975534 13 6.2374 ######
14 673736 975534 13 6.7374 ######
15 724289 975534 13 7.2429 #######
16 775600 975534 13 7.7560 #######
17 826843 975534 13 8.2684 ########
18 875950 975534 13 8.7595 ########
19 923411 975534 13 9.2341 #########
20 975534 975534 13 9.7553 #########
Here, the average roll is 13, a natural 20 is the statistically most common result, at just shy of 10%. The power is evident.
Disadvantage
DISADVANTAGE
ROLL CNT MAX AVG PERC BLOCK
1 975930 975930 7 9.7593 #########
2 924519 975930 7 9.2452 #########
3 876680 975930 7 8.7668 ########
4 825330 975930 7 8.2533 ########
5 774850 975930 7 7.7485 #######
6 724971 975930 7 7.2497 #######
7 674952 975930 7 6.7495 ######
8 623987 975930 7 6.2399 ######
9 574753 975930 7 5.7475 #####
10 524760 975930 7 5.2476 #####
11 475416 975930 7 4.7542 ####
12 425107 975930 7 4.2511 ####
13 375485 975930 7 3.7548 ###
14 324773 975930 7 3.2477 ###
15 274635 975930 7 2.7464 ##
16 224497 975930 7 2.2450 ##
17 174780 975930 7 1.7478 #
18 124667 975930 7 1.2467 #
19 74897 975930 7 0.7490
20 25011 975930 7 0.2501
And here, it looks like the opposite, which is understandable, because it is. The average is 7, or three below 10, just as the average for Advantage is three above. The statistically most common result is a natural 1, just shy of 10%.
Quad Disadvantage
At one point in Dimension 20’s Fantasy High: The Sophomore Year, Fig was doing something antisocial and annoying Brennan, so he said, for a homebrew rule, this was going to take a Quad Disadvantage roll: four rolls and you take the worst. Fig made it — she has crazy-good modifiers — but that was tough. I was redoing the analysis code, so I thought it would be easy to this.
QUAD DISADVANTAGE
ROLL CNT MAX AVG PERC BLOCK
1 1855748 1855748 4 18.5575 ##################
2 1586842 1855748 4 15.8684 ###############
3 1341491 1855748 4 13.4149 #############
4 1124391 1855748 4 11.2439 ###########
5 932683 1855748 4 9.3268 #########
6 761891 1855748 4 7.6189 #######
7 615093 1855748 4 6.1509 ######
8 488757 1855748 4 4.8876 ####
9 379663 1855748 4 3.7966 ###
10 288366 1855748 4 2.8837 ##
11 214185 1855748 4 2.1418 ##
12 153834 1855748 4 1.5383 #
13 106707 1855748 4 1.0671 #
14 69248 1855748 4 0.6925
15 42045 1855748 4 0.4204
16 23109 1855748 4 0.2311
17 10834 1855748 4 0.1083
18 4064 1855748 4 0.0406
19 978 1855748 4 0.0098
20 71 1855748 4 0.0007
Here, we’re getting close to a 20% chance of a 1, and an average roll of four. This is very “don’t go there” energy. If your DM starts talking about quad disadvantage, start thinking about changing your behavior.
Show Me The Code!
#!/usr/bin/env perl
use strict;
use warnings;
use feature qw{ say signatures state };
no warnings qw{ experimental };
use List::Util qw{ min max sum };
my $x = {};
my $y = {};
my $z = {};
my $q = {};
for my $i ( 1 .. 20 ) {
$x->{$i} = 0;
$y->{$i} = 0;
$z->{$i} = 0;
$q->{$i} = 0;
}
my $loop = 10_000_000;
for my $i ( 1 .. $loop ) {
$x->{ d20() }++;
$y->{ max( d20(), d20() ) }++;
$z->{ min( d20(), d20() ) }++;
$q->{ min( d20(), d20(), d20(), d20() ) }++;
}
analyze( 'normal', $loop, $x );
analyze( 'advantage', $loop, $y );
analyze( 'disadvantage', $loop, $z );
analyze( 'quad disadvantage', $loop, $q );
exit;
sub analyze ( $tag, $loop, $hashref ) {
say uc $tag;
say join "\t", map { uc } '', 'roll', 'cnt', 'max', 'avg', 'perc',
'block';
my $sum = sum map { $_ * $hashref->{$_} } 1 .. 20;
my $avg = int $sum / $loop;
my $max = max values $hashref->%*;
for my $i ( 1 .. 20 ) {
my $c = $hashref->{$i};
my $p = percent( $c, $loop );
my $b = blocks( $c, $loop );
say join "\t", '', $i, $c, $max, $avg, $p, $b;
}
say '';
}
sub d20 { return 1 + int rand 20 }
sub blocks ( $num, $denom ) {
return '#' x int( ( 100 * $num ) / $denom );
}
sub percent ( $num, $denom ) {
return sprintf '%.4f', ( 100 * $num ) / $denom;
}