Since FrogReaver has't replied, allow me to chime in.
We were comparing the expected damage of a Fighter vs. Paladin. His post #171 (top of page 18) show a table he made. We are now discussing how much of a greater impact the Battle Master using precision attack instead of one of the maneuvers that simply add damage. That is why he replied to my post about calculating the value of precision attack in this scenario and mentioned you if you want to do it.
I think that about covers it.
Ah, ok, thanks. Yeah, [MENTION=6795602]FrogReaver[/MENTION] and I had a very in the weeds discussion about that recently.
Since there seems to be no hope of a new thread, here's how I think about precision attack in a nutshell.
Suppose you need to roll a natural 9 to hit. The simplest approach is to assume you use a precision die any time you're within x of a hit (where x is the size of the die). So, if you're of a level where your superiority dice are d8s, then you'd use a die on a roll between 2 and 8 (1 is an auto-miss). With no advantage, rolls fall in this range 7/20 of the time, so you're averaging about one die every three attack rolls.
Okay, so how effective is each die at creating an extra hit? Well, if you're off by 1 it's effective 8 times out of 8. Off by 2 it's 7 out of 8, etc., on down to 2 out of 8 if you're off by 7. Each of these 7 possibilities is equally likely if we don't have advantage, so the chance that we get an extra hit out of rolling a die is:
1/7 * 8/8 + 1/7 * 7/8 + ... + 1/7 * 2/8
In this example, that's 0.625; that is each die is worth an extra 5/8 of an attack. If we're a greatsword-wielding fighter with 18 Strength, our attacks average 11 damage, so that means each precision attack die is worth 5/8 * 11 = 6.875 damage. Well over simply adding d8 damage to an attack, which would be worth 4.5 (which makes sense, since the maneuvers that add the die to the damage also have another effect). If our chance to hit is less than the 5/8, the precision die is worth even more damage than getting a whole extra attack attempt (from riposte, for example).
We can improve this even further if we are more conservative with our dice; maybe we don't risk wasting it if we're off by more than 5. In that case, each time we roll, we're off by somewhere between 1 and 5, with between an 8/8 and a 4/8 chance of converting the miss. If no advantage, those are equally likely, so each die has a:
1/5 * 8/8 + 1/5 * 7/8 + ... + 1/5 * 4/8 = 0.75
chance of giving us an extra hit. For the greatsword wielder that's worth 8.25 damage, more than adding d8 to damage can possibly do, and usually more than getting an extra attack attempt with riposte. And of course, if we're using the GWM power attack for example, we do even more damage on a hit, so the value of precision attack goes up even more.
However, now we're only going to roll the die 5/20 of the time, or one in every four attacks, which means if we're not getting twenty or so attacks between short rests, there's a decent chance we're going to get to a rest with unspent dice.
At the other extreme we could be super-conservative and only use our dice when we're within 1, so they always make the conversion, but unless it's a day with long combat stretches, we're very unlikely to use all our dic
If you make an assumption about how many combat turns you're likely to have between rests, you can calculate for each triggering threshold how likely you are to use all your dice, waste one, waste 2, etc. Obviously if you're level 3 and you're only making one attack per round, you want to be pretty liberal; probably just rolling any time it has a chance of working. But as you get more attacks, it makes sense to be more conservative, both to stretch out your dice through the day, and to get more value from each one. At any given point you can strike a balance between getting more value out of each die and having less chance of leaving dice on the table, so to speak.
Something counter-intuitive about this (it was counter-intuitive to me, at least, before doing the math) is the effect that advantage has on how useful precision dice are. You might think that if you already have advantage you get less value from precision attack, because you already have a better chance to hit. But actually the opposite is often true: Because higher rolls are more likely when you have advantage, for any given to-hit threshold, you're more likely to miss by a little than to miss by a lot. So that shifts the weights in calculating the conversion rate of a die toward the higher probabilities, giving you a better success chance per die.
If we need a natural 9 to hit as in the previous example, and we use a die any time we roll between 4 and 8, without advantage the number of extra hits per die was
1/5 * 8/8 + 1/5 * 7/8 + ... + 1/5 * 4/8 = 0.75
But with advantage, it's no longer 1/5 on each possibility; it turns out to be
0.27 * 8/8 + 0.24 * 7/8 + 0.20 * 6/8 + 0.16 * 5/8 + 0.13 * 4/8 = 0.80
So that each die is worth 8.8. However, a caveat is that with a relatively easy target like this (where a natural 9 hits), we will have fewer opportunities to use our dice if we stick to rolls that are within 5. Instead of coming up one time in four, this situation comes up less than one time in seven. So we may need to be more liberal with our dice to avoid having left overs, which will cut down some on the value per die.
But for a GWM or SS character where we're taking penalties to hit, or if we're facing really tough enemies, we can get the best of both worlds: being choosier with our dice and having less chance of wasting them. If we need a natural 14 to hit, we have a slightly more than 1/4 chance of missing by 5 or less, and more uses will be smaller misses, so we'll get about 0.77 extra hits per die and go through them slightly faster than if we didn't have advantage (therefore, with less chance of wasting any). So rather than diminishing returns when we combine advantage and precision attack, we actually get a synergy between them.