[RPG] How does Halfling Luck (re-rolling the Nat 1’s) affect the dice outcome


The Halfling Luck racial trait lets you reroll natural 1's on checks, saves, and attack rolls, but you must keep the second result. With Advantage or Disadvantage, it lets you re-roll only one of the dice if both turned out to be 1's.

How does this affect the expected statistical outcome of the roll? Note, also consider Advantage & Disadvantage, as the trait affects that roll as well.

Best Answer

It adds up to a +0.5 to all rolls

The average roll of a die is 10.5, which is just the average value of the numbers 1-20.

We re-roll any die that comes up as 1, and since we stick with whatever that outcome is, we can just replace the 1 in the list with the average roll again, and recalculate the average, giving us an average roll of 10.975.

Therefore, the halfling luck trait is approximately a +0.5 to all d20 rolls.

This is the distribution, as generated from a quick and dirty matlab script I wrote. As you can expect, it's pretty uniform for all values that are not 1 (note that there's still some variation, because it comes from actual random numbers and not an analytic calculation):

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Advantage and Disadvantage

I modified my script to see how it would affect advantage and disadvantage.

For disadvantage:

  • Normal average: 7.17
  • With Halfling luck:7.80

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For Advantage:

  • Normal average: 13.82
  • With Halfling luck: 14.13

enter image description here

We can see that there's about a +0.6 bonus for disadvantage, whereas there's only a +0.3 bonus for advantage. Of course, this makes sense, because rolling a 1 is more impactful for disadvantage than it is for advantage, as the plots make clear.

Also worth noting is that the fact that you only reroll one die doesn't really seem to change the average value. Again, this makes sense because you only have a 1/400 chance of rolling two ones.