Exploding Dice Averages

[WhosDaDungeonMaster]

Thank you for presenting the math in a more easily functional format.

I'm surprised the increase in average damage isn't that large.

I had been toying with the notion of replacing the natural 20 crit with exploding weapon damage, and people have warned me that the difference would be quite significant because the maximum value of a damage die will pop up more often than a natural 20.

Armed with this knowledge I might just attempt to swap natural 20 crits with exploding damage dice.

It could be quite a bit depending on what you need to hit on the attack roll.

Take the d6 above, for instance. With a natural 20 critical adding on average 3.5 damage, your expected increase for a d6 attack is 0.175 (3.5 * 1/20).

Think about the exploding d6 with an average of 4.2. This is an increase of 0.7 (4.2 - 3.5), but that assumes you are always hitting.
If you hit only on a 16 - 20 (0.25), then the expected increase is the same, that is 0.175 (0.7 x 0.25).
If you need 17 or higher, the exploding dice results in less of an increase.
However, if you hit on a 15 or lower, the exploding dice method would result in higher critical damage, on average.
Say, if a 11 or higher is needed, the expected increase is 0.35 (0.7 x 0.5), or double the natural 20 critical.

If the higher and more often critical damage doesn't bother you, go for it, as it isn't a bad idea. For me, however, it would slow things down because people would be rolling 6's, 8's, 10's, etc. for the exploding die damage than they would the natural 20.

 

[Yunru]

You would also need to adjust the n/(n-x) part. Rerolling 1's and 2's like with GWM also increases your chance of landing on one of the exploding numbers. So you would just need to do 1/(1-x) where x is your chance of landing on an exploding number with rerorolling 1's and 2's taken into account.

Wouldn't just lowering the number of sides but keeping the average the same work?

A d6 that rerolls all 1s and 2s is just a d4 with an average of 4.5, no?

 

[FrogReaver]

Wouldn't just lowering the number of sides but keeping the average the same work?

A d6 that rerolls all 1s and 2s is just a d4 with an average of 4.5, no?

Only if you reroll every 1 and 2 that comes up no matter what. I assume the question about reroll 1's and 2's was related to GWM. If so then doing what you propose won't work.

 

[MechaPilot]

It could be quite a bit depending on what you need to hit on the attack roll.

Take the d6 above, for instance. With a natural 20 critical adding on average 3.5 damage, your expected increase for a d6 attack is 0.175 (3.5 * 1/20).

Think about the exploding d6 with an average of 4.2. This is an increase of 0.7 (4.2 - 3.5), but that assumes you are always hitting.
If you hit only on a 16 - 20 (0.25), then the expected increase is the same, that is 0.175 (0.7 x 0.25).
If you need 17 or higher, the exploding dice results in less of an increase.
However, if you hit on a 15 or lower, the exploding dice method would result in higher critical damage, on average.
Say, if a 11 or higher is needed, the expected increase is 0.35 (0.7 x 0.5), or double the natural 20 critical.

If the higher and more often critical damage doesn't bother you, go for it, as it isn't a bad idea. For me, however, it would slow things down because people would be rolling 6's, 8's, 10's, etc. for the exploding die damage than they would the natural 20.

I generally assume that players hit 60% of the time. So, this would be the average d6 damage per attack of a player:

1-8 roll = miss = 0 damage (40% weight)
9-19 = non-crit hit = 3.5 damage (55% weight)
20 = crit = 7 damage (5% weight)
Average per attack of 2.275

1-8 roll = miss = 0 damage (40% weight)
9-20 = hit = 4.2 damage (60% weight)
Average per attack of 2.52

That's an average increase of 0.245

 

[WhosDaDungeonMaster]

Sure, the math works out the same: 0.7 x 0.6 - 0.175 = 0.42 - 0.175 = 0.245.

So, you just have to decide if getting an extra 0.245 points of damage on average for every attack, and having to roll the additional dice, would slow things down appreciably. I don't think it would, but of course, it would hurt the PCs just as quickly.

 

[MechaPilot]

Sure, the math works out the same: 0.7 x 0.6 - 0.175 = 0.42 - 0.175 = 0.245.

So, you just have to decide if getting an extra 0.245 points of damage on average for every attack, and having to roll the additional dice, would slow things down appreciably. I don't think it would, but of course, it would hurt the PCs just as quickly.

True.

I do think it has a sort of immersion benefit to it as well. With exploding damage dice, every lethal weapon can be one-hit lethal, which makes sense for lethal weapons. The chances of that happening are reduced as the foe's HPs increase, but it is, in theory, possible to kill a dragon (or a PC) with a single longbow shot . . . if you roll enough consecutive 8s.

I will have to figure out what to do with magic weapons though. At my table I don't give out +X weapons and armor. Instead of having a +X, weapons have expanded crit ranges. Instead of having a +X, armors impose expanded crit fail ranges on attackers. I suppose I could have the expanded crit range result in an expanded explosion range.

So, instead of having a d8 weapon that crits on a 19-20, which according to my math is roughly analogous to a d8 +1 weapon, that weapon's damage would explode on a 7 or 8.

 

[WhosDaDungeonMaster]

Here is the table showing you the mean damage by die type and explode on number. You can see the percentage increase next to it.

For example, a +3 d8 weapon (by your idea, would explode on 5, 6, 7, or 8) would have a mean damage of 9, or a 100% increase above the otherwise mean of 4.5.

Just for you to think about.

 

[MechaPilot]

Here is the table showing you the mean damage by die type and explode on number. You can see the percentage increase next to it.

For example, a +3 d8 weapon (by your idea, would explode on 5, 6, 7, or 8) would have a mean damage of 9, or a 100% increase above the otherwise mean of 4.5.

Just for you to think about.

Thank you for pointing that out, and especially for the table.

Hmm. That means I'll have to think of something else to do about magic weapons.

I suppose I could go back to +X weapons and armors. It makes figuring out encounter difficulty a tad bit more work. That's not such a big problem, but I really don't care for static bonuses that fade into the background, that makes magic items feel boring to me.

 

[WhosDaDungeonMaster]

Thank you for pointing that out, and especially for the table.

No problem. Glad it helped. I was intrigued by the idea, so wanted to look into it further.

It would be pretty crazy to have a Shortsword +3 dealing 10.5 damage per hit on average. LOL!

EDIT:

Here is an option for magical weapons (but it is a bit more complex than a straight modifier...):

Find the base damage (or the closest, using d12 for 2d6 weapons). For each +1 of the weapon, move down the table one level.

[TABLE="width: 256"]
[TR]
[TD="class: xl76, width: 64, align: center"]Base[/TD]
[TD="class: xl71, width: 64, align: center"]Min[/TD]
[TD="class: xl72, width: 64, align: center"]Max[/TD]
[TD="class: xl73, width: 64, align: center"]Avg[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d4[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]4[/TD]
[TD="class: xl67, align: center"]2.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d6[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]6[/TD]
[TD="class: xl67, align: center"]3.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d8[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]8[/TD]
[TD="class: xl67, align: center"]4.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d10[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]10[/TD]
[TD="class: xl67, align: center"]5.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d12[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]12[/TD]
[TD="class: xl67, align: center"]6.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d6+d8[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]14[/TD]
[TD="class: xl67, align: center"]8[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]2d8[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]16[/TD]
[TD="class: xl67, align: center"]9[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d8+d10[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]18[/TD]
[TD="class: xl67, align: center"]10[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]2d10[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]20[/TD]
[TD="class: xl67, align: center"]11[/TD]
[/TR]
[TR]
[TD="class: xl75, align: center"]2d12[/TD]
[TD="class: xl68, align: center"]2[/TD]
[TD="class: xl69, align: center"]24[/TD]
[TD="class: xl70, align: center"]13[/TD]
[/TR]
[/TABLE]

For example, a Longsword +2 would deal d12 damage instead of 1d8+2. The average is the same (6.5), but it allows for higher damage potential. For most cases, the average is the same or only slightly more using this concept.

For magical armors, you could rule instead each +1 grants one point of magical DR. This would be more powerful, essentially, but not overly so depending on how common magic armors are in your game.

 

[TheCosmicKid]

Here is an option for magical weapons (but it is a bit more complex than a straight modifier...):

Find the base damage (or the closest, using d12 for 2d6 weapons). For each +1 of the weapon, move down the table one level.

[TABLE="width: 256"]
[TR]
[TD="class: xl76, width: 64, align: center"]Base[/TD]
[TD="class: xl71, width: 64, align: center"]Min[/TD]
[TD="class: xl72, width: 64, align: center"]Max[/TD]
[TD="class: xl73, width: 64, align: center"]Avg[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d4[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]4[/TD]
[TD="class: xl67, align: center"]2.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d6[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]6[/TD]
[TD="class: xl67, align: center"]3.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d8[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]8[/TD]
[TD="class: xl67, align: center"]4.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d10[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]10[/TD]
[TD="class: xl67, align: center"]5.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d12[/TD]
[TD="class: xl65, align: center"]1[/TD]
[TD="class: xl66, align: center"]12[/TD]
[TD="class: xl67, align: center"]6.5[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d6+d8[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]14[/TD]
[TD="class: xl67, align: center"]8[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]2d8[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]16[/TD]
[TD="class: xl67, align: center"]9[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]d8+d10[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]18[/TD]
[TD="class: xl67, align: center"]10[/TD]
[/TR]
[TR]
[TD="class: xl74, align: center"]2d10[/TD]
[TD="class: xl65, align: center"]2[/TD]
[TD="class: xl66, align: center"]20[/TD]
[TD="class: xl67, align: center"]11[/TD]
[/TR]
[TR]
[TD="class: xl75, align: center"]2d12[/TD]
[TD="class: xl68, align: center"]2[/TD]
[TD="class: xl69, align: center"]24[/TD]
[TD="class: xl70, align: center"]13[/TD]
[/TR]
[/TABLE]

For example, a Longsword +2 would deal d12 damage instead of 1d8+2. The average is the same (6.5), but it allows for higher damage potential. For most cases, the average is the same or only slightly more using this concept.

For magical armors, you could rule instead each +1 grants one point of magical DR. This would be more powerful, essentially, but not overly so depending on how common magic armors are in your game.

I don't think this option goes well with exploding dice. Increasing die size decreases the probability of an explosion. Expected damage still increases, but as a gaming experience, it "feels bad". And then, suddenly, when you jump from d12 to d6+d8, the probability shoots up again, and the overall distribution shifts from flat to normal. It's weird.