Another issue (and it's a big one) that we can't answer is how many times you can use your dice. If you can choose to use your Inspiration after your roll and you know what you missed by (something I highly encourage as I think it's more fun that way) then you can, in principle, guarantee success by adding a d6.
Miss by one? You now automatically succeed! If you have lots of rolls and few uses or only one use in a session with a dozen or two rolls it can pay wait for that roll you miss by one or two and guarantee or almost guarantee a success. Though in this case you don't really get to choose whether that roll was an important roll or not. Rolling another d20 means that the success of the second roll is not tied in any way whatsoever to the value of the first roll (beyond the first roll being a failure). So it can be used just as successfully on the incredibly important roll you missed by 10.
I'd still rather have another d6, though.
I did account for the 'miss by 1' with the +d6 effect, though. It happens 5% of all rolls and is 100% effective, or 5% overall effective. The extra d20
still is better in almost every case. Here's a new chart, where I'm showing the difference in effectiveness between the d20 reroll and the +d6 total effect for each target number 1-20. Blue is where the d20 reroll is more effective, orange is where the +d6 is more effective. The number at the top of the bar is how much the chance is increased to hit that number
over the other method (so, not raw increase, but increase relative to the other method).
As you can see, the d20 reroll is much more effective from 3 to 16 than the +d6 (accounting for miss by 1's) while the +d6 is more effective from 19 and 20. At 2, 17, and 18, they are very close in effectiveness.
For those interested, you figure the effectiveness of the d6 by determining the likelihood of rolling 1 less, 2 less, ..., 6 less than the target number. That's easy for a d20, though, is 1/20 for each. Then you figure for each X less what the odds are for rolling high enough on the d6 to turn it into a success. That's pretty easy too, as it 6/6 for 1 less, 5/6 for 2 less, ..., and 1/6 for 6 less. You then multiple the odds of rolling X less times the odds the d6 turns it into a success for each X less and then sum. So:
1 - (1/20 * 6/6) + (1/20 * 5/6) + ... + (1/20 * 1/6) =
You can simplify this by factoring out the 1/20 and getting:
2 -- 1/20 * (6/6 + 5/6+ 4/6 + 3/6 + 2/6 + 1/6) =
Summing, it's:
3 -- 1/20 * 21/6 = 21/120 = 0.175 = 17.5%
That's the average increase for +d6. To get the actual effectiveness, you have to look at the odds of failing for each d20 roll (as you won't get a benefit if the d20 is a success). For example, if you need a 10, then you'll only fail 9/20 times (45%), so the 17.5% bump the d6 is only useful 45% of the time, or (45% * 17.5%) = 7.9% average increase in success.
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I like these exercises not because I like min-maxing, but because you get to see what the mechanic actually incentivizes. Knowing the mechanical impact is one part of that. The second is the feel at the table. I think it's good to know that the +d6 method provides for worse outcomes than the d20 reroll over the majority of the instances (and will, in fact, be most likely useful to non-proficient, high difficulty rolls). However, there's still the preference at the table, the 'feel' of the mechanic, and some may prefer the more limited variance of the d6 to the larger variance of the d20. If you, as a player, really focus on those 'miss by 1' moments -- ie, you really want your hero point mechanic to come through reliably in the narrow cases, then the d6 is of more value to you aesthetically because it does work more reliably in those cases (missing by one is a 1/20 chance, but, once you're already there, the 100% conversion of the d6 is nice vs the lower conversion of the d20 reroll), even if it's less useful over the broader cases of all d20 roll in a session. The reroll is much more likely to be useful on crux rolls than the d6, as the use curve is much broader, but also more likely to fail at 'miss by 1' situations. And that preference is valid. It's good to know the numbers, though, if you're evaluating the mechanic.
