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In D&D 5th Edition the Sorcerer Metamagic ability Empowered Spell states the following:

"When you roll damage for a spell, you can spend 1 sorcery point to reroll a number of the damage dice up to your Charisma modifier (minimum of one). You must use the new rolls."

How do you calculate the increase in the average damage when you use this ability? It seems to me you would use this to only reroll dice that came up 3 or less however you have less of a chance of increasing your damage when you reroll a 3 than when you reroll a 1.

I know something of probability from playing tabletops for so long but figuring this out is beyond me.

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It's complicated

While skoormit has pointed out that the average percent increase for 1 die is 21.43, the overall increase for multiple dice is not straightforward, and highly depends on your Charisma modifier, more so for more dice.

I used a small python script to roll xd6. The lowest c dice (where c = positive Cha modifier) were re-rolled, unless there were less than c dice below average. The following data was gathered by averaging over 100,000 of such sets for each combination of x and c.

It shows the percentile increase of the average roll, i.e. ((new avg / old avg) − 1) × 100.

graph of five curves for the expected percentage increase in damage by number of dice rolled, for five values of Cha from +1 to +5

As you can see, the increase drops rapidly below 20% for x > c, though the slope becomes less steep as c increases.

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Simple Case: Available Rerolls Equal to Number of Dice Rolled

You can expect half of the dice you roll to be below average. You will reroll those dice.

For a d6, you are going to reroll the 1s, 2s, and 3s. Those dice (1, 2, and 3) have an average value of 2. The reroll has an average value of 3.5.

So, half of your dice will increase by 1.5, on average. That's an effective increase of 0.75 per d6 that you roll.

average result of a d6 with no reroll allowed: 3.5

average result of a d6 with a reroll allowed: 4.25

In other words, the reroll adds ~21.4% to your average result when rolling a d6.

For a die with n sides, this is the formula for average results with and without a reroll:

without a reroll: (n+1)/2

with a reroll: (5n/8) + 0.5

A little bit of algebra (not included here) tells us this:

(Average result without a reroll) * ( 1 + n / ( (n+1) * 4 ) ) = (Average result with a reroll)

In english: Divide n by n + 1, then divide that by 4. Multiply by 100, and that's the percentage increase you can expect with a reroll.

Percent increase for common dice (sides: pct increase)--

  • 4: 20.00%
  • 6: 21.43%
  • 8: 22.22%
  • 10: 22.73%
  • 12: 23.08%
  • 20: 23.81%

Complex Case Estimate: Available Rerolls Less Than Number of Dice Rolled

For d6s, if you have fewer rerolls available than the number of dice in your damage roll, just multiply the number of dice by 3.5, then add 0.75 for each reroll available. That will give you a conservative (slightly low) estimate of your average result.

For example, if you are rolling 6d6 and have 3 rerolls:

6 * 3.5 = 21 ; 3 * 0.75 = 2.25

21 + 2.25 = 23.25

Your conservative (slightly low) estimate for your average result is 23.25 (which is 10.7% better than without the rerolls).

This estimate is slightly low because you get to choose which dice you reroll, and if you have more below-average rolls to choose from than you have dice to reroll, you will always reroll the lowest dice.

For an example consider the case of rolling 2d6 with 1 reroll available.

Our estimate comes up with: ((2 * 3.5) + (1 * 0.75)) = 7.75 However, if you take all 36 possible outcomes and trade out the lower below-average die (if two dice are 3 or less) for a 3.5, the average result is about 8.24. (Details are here.)

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  • \$\begingroup\$ That assumes that you always reroll all results below the average. While this will pay out on average, from time to time, you may be better off not taking risks and accepting a slightly below-average result. \$\endgroup\$ – Strill Oct 5 '14 at 3:09
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    \$\begingroup\$ @Strill that should be irrelevant as long as you're counting the second roll as 3.5. Yes it happens, but that's something that is factored in when you consider the average. \$\endgroup\$ – wax eagle Oct 5 '14 at 3:44
  • \$\begingroup\$ Yeah you can consider the average to be 3.5, but that's an average over an infinite number of trials. In practice, there's a good chance you'll get worse results. If that little bit of damage might make the difference, there's a good argument to be made for keeping rolls that are just below the average. Once you decide that you won't necessarily always reroll below-average results, then it changes the calculations. \$\endgroup\$ – Strill Oct 5 '14 at 4:08
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    \$\begingroup\$ @Strill, I agree that sometimes it's best not to try to increase a roll that's only slightly below average. But the question is specifically asking about average damage. \$\endgroup\$ – skoormit Oct 6 '14 at 13:47
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When you reroll a die, you can statistically expect to get an average roll. Therefore, you compare what you got to what the average is, and the difference is how much more damage you'll probably do.

The average roll for any die is [sides / 2] + 0.5. For example, a d10 is 10/2 + 0.5 = 5.5.

So if you rolled a 1 on a d10, you can expect a 5.5 on your reroll, for 5.5 - 1 = +4.5 damage on average

If you rolled a 2, you can expect 5.5 - 2 = +3.5 damage increase.

If you rolled an 8, however, you can expect 5.5 - 8 = -2.5 damage DECREASE.

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So we have to ask ourselves, how likely are we to need a reroll, how many rerolls do we need, and then, how much does this increase our damage output. I'm going to use a d6 for my examples, because, frankly, it's the most common spell die and the one you're most likely to be concerned about. Similar analysis can be done for higher dice values, although the result will be the same, the damage boost from ES is minimal.

First, the easy part, and the part that will make this whole thing tick. When is it a good idea to reroll. Obviously, rerolling a one is a good deal, you can't make a roll worse by rerolling it. The expected outcome of rerolling a d6 one once is

$$ \dfrac{5}{6} \times \dfrac{2 + 3 + 4 + 5 + 6}{5} + \dfrac{1}{6} \times 3.5 = 3.91 $$

So we definitely want to reroll 1s when we get them. Now is rerolling 2s a good idea?

$$ \dfrac{4}{6} \times \dfrac{3+4+5+6}{4} + \dfrac{2}{6} \times 3.5 = 4.16 $$

So we do even better when we reroll 1s and 2s. Let's check out 3s.

$$ \dfrac{3}{6} \times \dfrac{4+5+6}{3} + \dfrac{3}{6} \times 3.5 = 4.25 $$

This is obviously high risk, but it does lead to better outcomes on average. You should reroll 1s,2s and 3s if you can. Rerolling 4s gives:

$$ \dfrac{2}{6} \times \dfrac{5+6}{2} + \dfrac{4}{6} \times 3.5 = 4.16 $$

And rerolling 5s gives

$$ \dfrac{1}{6} * 6 + \dfrac{5}{6} * 3.5 = 3.92 $$

This means it's basically always better to take the reroll on average even on a 5 because the second chance at a 6 is pretty strong. However, the strongest roll is to reroll 1s, 2s and 3s. So we're going to stick with that as it represents a relatively conservative take on this. We will use 4.25 as the value of a rerolled die.

You get to reroll a number of dice equal to your charisma modifier, probably +3 for a low level sorcy, and up to 5. I'm going to use 4 in this example because I want an even number (and hell, it represents an average of 3, 4 and 5).

So if you use a power that rolls 4 or fewer dice, you gain an average of .75 per die of damage rolled when you invoke this and reroll 1s, 2s and 3s. Thus you can treat all the dice in that power as 4.25 when you calculate damage averages. So ultimately, on a power with four damage dice (The maximum in this scenario), Empowered spell will provide 3 extra damage on average (14 -> 17, about 20% more).

What about spells with more damage dice though?

Well, what we need to know is how many dice do we have to roll before we're using all of our rerolls on dice 1-3 (and then 1-2 and then just 1s). The reroll becomes less impactful as you roll more dice, but where is the sweet spot?

For a damage roll of more dice, we need to know how many dice are going to get rerolled. We're going to work two more expressions, and figure out what our average damage is.

The first thing we'll figure out is Fireball, the damage here is 8d6. There is a 64% chance that we'll get to reroll all of the 1-3 we roll, and a 91% chance we'll get to reroll all the 1s and 2s we roll (and a basically 100% chance we'll get to reroll all the 1s). Let's see what happens when we reroll the 1-3 dice. The four reroll dice are counted as we calculated the reroll dice above. The remaining dice are most likely to show 4,5 and 6. It's worth calculating the fifth dice, but the rest of the series converges to 5 and so we'll use that after that (the 6th die counds as 4.9, so we'll take 5 there).

$$ 4 \times 4.25 + .6367 \times 5 + .3643 \times 2 + 5 \times 3 = 35.91 $$

This is nearly and 8 point increase over normal damage and nearly 30% more.

Meteor Swarm on the other hand rolls 20d6 (and does so 4 times, I'm honestly not sure how that quite works, not sure if it's all 1 damage roll, or 4 seperate, would have to read into it a bit more, if this is 4 rerolls in 80d6, that makes it pointless, we'll use 20 here as a reasonable test of the scaling here), truthfully the odds of rolling even 4 1s is strong enough that we're going to consider that most likely that's all you'll get to reroll (this also serves to limit the series we calculate, which is mostly in the interst of time, running this calc for 1s and 2s should be something you do before making a final decision, but truthfully, its probably only marginally better than rerolling 1s since you're pretty likely to see 4 1s in a set of 20d6).

$$ 3.91 \times 4 + 1 \times .8982 \times 4 + 1 \times .1018 + 4 \times 15 = 79.33 $$

Which is a bit above 10% for increase on the normal damage for this power.

So the moral of the story here is that it's a solid increase until you hit L9 spells.

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As you say, it depends on what you rolled.

The average increase on each individual dice you reroll will be (Dice roll average value, I.E. 3.5 on a d6) - (current value of dice you are rerolling).

As an example, if you're rerolling 3d6, one of which came up 1, one of which came up 2, and one of which came up 3, you'll see an average increase of:

(3.5 - 1) + (3.5 - 2) + (3.5 - 3)

= (2.5) + (1.5) + (0.5)

= 4.5 damage.

If you're looking to 'pre-calculate' how much higher you can expect the damage of a spell you cast to be if you're expecting to empower it in advance for some reason, that's more complicated. Since you determine whether or not to empower after seeing the damage roll results, however, I'm not sure that's necessarily a relevant calculation.

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