In D&D 5th Edition, the Sorcerer's Metamagic option 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 metamagic option?
It seems to me you would use this to only reroll dice that had a result of 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 tabletop games for so long, but figuring this out is beyond me.
Best Answer
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.
As you can see, the increase drops rapidly below 20% for x > c, though the slope becomes less steep as c increases.