INTRODUCTION: I took the liberty of doing some math to illustrate why rolling with disadvantage is very, very punishing. Rolling with disadvantage is not the same as roll with a flat bonus or penalty. It's worse!
DISCUSSION: When you roll two dice, the number of possible combinations of numbers you can roll is equal to the highest value of each die multiplied together. Example: 20 x 20 = 400. On die #1, you can roll any number from 1 to 20, but if die #2 rolls a 1, you rolled a 1. Therefore, you have twenty combinations to roll a natural 1. But that's not all! The combinations rolled with die #1 being variable and die #2 being static flip when die #1 is static and die #2 is variable. To illustrate, on die #1, you can roll a 1, and on die #2, you can roll any number from 1 to 20 and still roll a natural 1. We throw out the second 1, 1 combination because it's a duplicate combo, and so we are left with thirty-nine possible combinations of rolling a natural 1 out of a total of 400 combinations of rolling two d20s! By contrast, there is only one combination to roll a natural 20 with disadvantage (20, 20), one combination out of 400!
The table found below illustrates the above.
Here's how to read this table if you're struggling to make sense of it. The column "Number Rolled" shows the lowest number on either d20, and so this is the roll we'll be accepting. The "Straight d20 roll percents" column is for comparison and shows the percentage chance to roll any individual number and a cumulative descending chance to roll that number or less. The column "D20 rolls w/Disadv. # Combinations and % Chance to Roll" shows the total number of combinations to roll that individual number, then the cumulative number of that number. The column "Disadv. Cumulative Descending" shows the chances to roll that number or less. In other words, read row "5" as 31 combinations to roll the number 5 with disadvantage, "173 of 400" is the number of combinations of die rolls that will give you a 5 or less out of all 400 combos, and the percentage of rolling a 5 is 7.75%. The lost column shows that one has a 64% chance to roll 5 or less.
| Number Rolled | Straight d20 roll percents | D20 rolls w/ Disadv. # Combinations and % Chance to roll # | Disadv. Cumulative Descending |
|---|---|---|---|
| 1 | 5% / 100% | 39 (39 of 400) = 9.75% | 100% |
| 2 | 5% / 95% | 37 (78 of 400) = 9.25% | 90.25% |
| 3 | 5% / 90% | 35 (105 of 400) = 8.75% | 81% |
| 4 | 5% / 85% | 33 (140 of 400) = 8.25% | 72.75% |
| 5 | 5% / 80% | 31 (173 of 400) = 7.75% | 64% |
| 6 | 5% / 75% | 29 (204 of 400) = 7.25% | 56.25% |
| 7 | 5% / 70% | 27 (231 of 400) = 6.75% | 49% |
| 8 | 5% / 65% | 25 (256 of 400) = 6.25% | 42.25% |
| 9 | 5% / 60% | 23 (279 of 400) = 5.75% | 36% |
| 10 | 5% / 55% | 21 (300 of 400) = 5.25% | 30.25% |
| 11 | 5% / 50% | 19 (319 of 400) = 4.75% | 25% |
| 12 | 5% / 45% | 17 (336 of 400) = 4.25% | 20.25% |
| 13 | 5% / 40% | 15 (351 of 400) = 3.75% | 16% |
| 14 | 5% / 35% | 13 (364 of 400) = 3.25% | 12.25% |
| 15 | 5% / 30% | 11 (375 of 400) = 2.75% | 9% |
| 16 | 5% / 25% | 9 (384 of 400) = 2.25% | 6.25% |
| 17 | 5% / 20% | 7 (391 of 400) = 1.75% | 4% |
| 18 | 5% / 15% | 5 (396 of 400) = 1.25% | 2.25% |
| 19 | 5% / 10% | 3 (399 of 400) = 0.75% | 1% |
| 20 | 5% / 5% | 1 (400 of 400) = 0.25% | 0.25% |
Here's what we can determine about rolling the d20 with disadvantage by analyzing this table. Most of our rolls are going to be 10 or less. Another way to phrase this is that most of our rolls are going to be the lower half of the range of numbers that we can roll. Nearly one quarter of your rolls will get you a 3 or less. Half of your rolls will get you a 6 or less. Three quarters of all rolls will get you a 10 or less. Your chances or rolling better than a 10 with disadvantage are exponentially crappier; for the purposes of emphasizing my point, that means that your chances of success get crappier and crappier and crappier.
Let's use a concrete example.
15 is a common Difficulty Challenge (DC) in D&D 5e for characters of lower levels. +5 is a common bonus for a skill if a first level character particularly excels in it. If we roll straight (meaning without advantage or disadvantage), the player must roll a 10 to succeed. With their +5, they meet 15. We have a 55% chance to roll a 10 or better when rolling straight. By imposing disadvantage, they go from a 55% chance to a 25% chance to roll a 10 or better. This is equivalent to a change in difficulty of 30%.
Any other character who does not have this +5 bonus will have a different and worse change in difficulty. If you had a +2 bonus instead, you have to roll a 13 to meet or beat 15. When rolling straight, you have a 65% chance to roll a 13 or better. With disadvantage, you have a 12.25% chance to roll a 13 or better. This means the change in difficulty is 52.75%! By having a slightly lower bonus, our odds change by a lot! This means that a +1 bonus is no longer worth a +5% as you would expect when rolling straight. The value of each +1 bonus or -1 penalty changes based on the face of the die rolled! In effect, your bonuses and penalties have a variable value. To reiterate, this is not consistent with the expectation that the advantage / disadvantage rule is equal to a flat bonus or penalty!
To be exhaustive, rolling two dice means we are no longer working with a linear distribution, we're working with a curve or slope. In the example above, a +5 bonus goes from being worth +25% when rolling straight to being worth +18.75% when rolling with disadvantage, and a +2 goes from being worth +10% when rolling straight to being worth +6 % with disadvantage. But slopes are not intuitive; the adjusted values given for disadvantage are for rolling the specific numbers for this specific case only! If the DC was 16, that +2 is still worth +10% when rolling straight, but with disadvantage its value becomes a +5%!
For the sake of clarity, and to exhaust all possible ways to think about this rule, consider that once upon a time, a Game Master might simply impose a flat circumstantial penalty to a roll of say -2 or -4. In the specific example above with the DC of 15, imposing disadvantage on the character with the +5 bonus is equal to imposing a -6 circumstantial penalty to the roll, but imposing disadvantage on the character with the +2 bonus is equal to imposing a -10 circumstantial penalty to the roll! That is absolutely punishing!
Let's do another example. Let's say I have no bonus or penalty, so a +0, and the DC is 15. A straight roll means I have a 30% chance to succeed. With disadvantage, I have a 6.25% chance to succeed. To be exhaustive, that means your chances of success with a +0 are nearly half that if you have a +2 and nearly one-quarter that if you have a +5. The be exhaustive, you have a 5% chance to roll a natural 20 when rolling straight. That means if I have a +0 and I have disadvantage on a DC 15 roll, my odds are only slightly better than rolling a nat 20 when rolling straight, but still worse than rolling a 19 when rolling straight.
Disadvantage is so punishing it's like the GM is practically saying F you and I refuse to play 5e. If the GM imposes disadvantage, you are going to fail. I want to reiterate this the second most: the math is now unintuitive! Do you see why I hate advantage / disadvantage with enough passion to write this!?
SUMMARY (what to tell your friends): When you roll one die, you are randomly generate numbers with a linear distribution, and so any +1 bonus or -1 penalty has a consistent value, and in the case of a d20, this is +/- 5%. When you roll two dice, you generate results on a slope, and so the value of any +1 bonus or -1 penalty becomes variable based on the number you roll on the die. The slope caused by disadvantage means your chances of rolling higher numbers is exponentially worse. Advantage / disadvantage therefore is not equivalent to adding a flat bonus or penalty to an individual roll. In fact, imposing disadvantage can easily be more punishing that imposing a massive circumstantial penalty such as a -5 or -10 to a d20 roll.
PROPOSED SOLUTION: Instead of advantage / disadvantage from D&D 5e, use EASY / HARD from Index Card RPG. EASY / HARD means that the target number (or DC) changes by +3 or -3. That's equal to a +15% or -15%. That's it. Happy gaming. Hankerin Ferinale is a genius and he should be king of D&D. The people in charge of 5e who came up with dis/advantage should be serving time.
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