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Under the Hood: Roll 2 Dice, Take The Highest

**Bedrockbrendan**:

BY RICK MOSCATELLO

Dungeons and Dragons 5E has a few “roll two dice, take the highest” mechanics. The most popular, for good reason, is the Advantage mechanic for rolling a D20 to hit, which is fairly close to a +4.

There are also a few ways to get a “roll two dice, take the highest” effect when rolling damage. Before looking at how this mechanic works for damage, let’s first take a quick look at how one figures the average damage for a single die.

When rolling a single die for damage, you take an expectation. For a D4, for example, you take 1/4 times 1 + (1/4) times 2 + (1/4) time 3 + (1/4) times 4…which adds up to an expected damage of 2.5. I could also just factor out the (1/4), and get (1/4) * (1 + 2 + 3 + 4), a slightly easier way to calculate.

Rather than do that kind of calculation, most players, when dealing damage, just add up the lowest and highest number on the die, and divide by two. That works for a single die

So, the average damage for a D6 is 3.5, (1 + 6)/2. This trick works for a single die roll, but not when rolling multiple dice or dealing with a a strange new mechanic, although taking an expectation still works.

Let’s take a look at expected damage with this mechanic for various dice:

Roll 2 D4, take the highest

Let’s consider the four possible values, and the number of ways you can get those values—each way is equally likely amongst the 16 possible outcomes from rolling a D4 twice.

1 damage is achieved in 1 way, rolling (1,1)

2 damage is achieved in 3 ways, rolling (1,2), (2,1), or (2,2)

3 damage is achieved in 5 ways, rolling (1,3), (2,3), (3,2), (3,1) or (3,3)

4 damage is achieved in 7 ways, rolling (1,4), (2,4), (3,4), (4,3), (4,2) (4,1) or (4,4)

Note how the number of ways of rolling a damage number increases by two each time the damage increases by one—this pattern works for larger dice as well.

Note also how when I add up all the ways, I get 16—I’ve accounted for all the possible outcomes of rolling a D4, two times.

So now I take expectation.

(1/16) * ( 1 * 1 + 3 *2 + 5*3 + 4 *7) = 3.125.

Thus the new mechanic isn’t much of an improvement for a D 4—you basically expect to do an extra 0.6125 point of damage per attack. Whoop de do.

Roll D6 twice, take the highest

Again, I set up the table of all possible damage rolls, and number of ways I can get that damage (I’ll not bother listing the actual events):

1 damage in 1 way

2 damage in 3 ways

3 damage in 5 ways

4 damage in 7 ways

5 damage in 9 ways

6 damage in 11 ways

Note the number of ways adds up to 36—it’s a cutesy trick. Anyway, I take expectation:

(1/36) * ( 1 * 1 + 2 * 3 + 3 * 5 + 4 * 7 + 5 * 9 + 6 * 11) = 4.47

Gee whiz, the new mechanic is almost identical to just adding 1 point of damage (since the expected roll of a D6 is 3.5).

Roll D8 twice, take the highest

The great thing about seeing the pattern is it really speed up calculating:

1 damage in 1 way

2 damage in 3 ways

3 damage in 5 ways

4 damage in 7 ways

5 damage in 9 ways

6 damage in 11 ways

7 damage in 13 ways

8 damage in 15 ways

With 64 possible outcomes of rolling a D8 two times, it’s good to have a trick. Anyway, the expected damage is 5.8125—well over 1 extra point of damage now.

It’s very clear that as the die gets bigger, the new mechanic becomes more powerful.

When we get to “D10, take the highest”, the expected damage is 7.15—1.65 extra damage. Nice, but hardly game breaking.

Finally, with “D12, take the highest” the expected damage is 8.49—almost two whole extra damage! Woohoo!

While nobody rolls a D20 for damage (I had monsters rolling it in my 15th level 4e game, though), the new mechanic would give an expected damage of 13.825, well over 2 points. One might have guessed it would be closer to 4 points, since Advantage is basically a +4 to hit, but the usefulness of a bonus varies with how hard it is to hit, unlike damage which is always just, well, damage. Note that Advantage for saving throws is thus more like a +2.

While not the most amazingly powerful mechanic from a game standpoint, it is at least one less modifier to add to a single die roll, and getting the most out of it hardly requires any game mastery—everyone knows when its time to bash monsters, you always want to roll the biggest die you can grab, after all.

Edit: The average roll of a D20 is 10.5. This makes the 3.325 bonus from the "roll two dice, take the highest" mechanic more in line with what was found in the Under the Hood article looking at Advantage in 5e. The author apologizes for the error, and notes that this should be taken as evidence that "rolling a 1" on a skill check for even something simple is still quite obviously a failure.

**Bren**:

Nice explanation Brendan. I've been interested in the Bonus/Advantage system as it is one of the few new (to me) ideas I've seen in a while.

Honor+Intrigue uses a Bonus/Penalty die system that is similar. There are three added features of Bonus and Penalty dice that your article didn't mention.

1) Adding a Bonus or Advantage dice doesn't change the maximum or minimum result. This means that unlike a flat bonus there is no additive effect to the maximum. While the average damage increases the best and worst results don't change.

2) If you have a simple critical/autohit and fumble/automiss system where critical is the max on the die and the fumble is the min on the die, e.g. for 1D20, a natural 20 is a critical and natural 1 is a fumble. Then by using a Bonus or Advantage die you are also increasing the probability of getting a critical (and decreasing the probability of getting a fumble). Which intuitively makes sense if you have an Advantage.

Note that this effect does not occur with a flat +2 type bonus since the chance of rolling a natural 20 is unchanged. and a fumble is the min onhen using a Bonus.

3) You can also use a Penalty die to simulate a Disadvantage. In this case you roll two dice and take the worst of the two results. Calculating the result follows the same methodology as in the OP. Note that just as with an Bonus or Advantage die, the maximum and minimum don't change. However the chance for a critical or autosuccess decreases and the chance for a fumble or autofailure increases. Which again accords with an intuitive understanding of "Disadvantage."

**Bedrockbrendan**:

Just want to clarify something: most of these articles are written by other people, I did not write this one. Unless the article says 'BY BRENDAN DAVIS' it is not mine. Each article has the author's name in the heading and I repeat it at the start of the body text.

**Bren**:

--- Quote from: BedrockBrendan;794852 ---Just want to clarify something: most of these articles are written by other people, I did not write this one. Unless the article says 'BY BRENDAN DAVIS' it is not mine. Each article has the author's name in the heading and I repeat it at the start of the body text.

--- End quote ---

Thanks for clarifying Brendan. Clearly I am too easily confused by screen names, pen names, and real names. :)

**rawma**:

13.825 is 3.325 over the average of 10.5 for d20, so it is closer to +4 than to +2.

The formula in general for dN is sum(i*(2i-1))/N^2, where i ranges from 1 to N; this simplifies to (2*sum(i^2)-sum(i))/N^2, which is (2*(N/6)*(N+1)*(2N+1)-(N/2)*(N+1))/N^2; which in turn is (N+1)*(2/3N-1/6)/N versus the normal average of N/2+1/2. Finally, the difference is (N/2+1/2)*(1/3N-1/3)/N, so the ratio of advantage roll average to regular average is 1/3-1/(3N).

The major appeal of advantage/disadvantage in 5e is that it doesn't stack; having both (even in unbalanced amounts) cancels, and having multiple advantage does nothing more. The change in critical/fumble chance is also an improvement.

(Post #50, and it's math. Life continues to be good.)

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