Assume you have a Fighter/Rogue with one Extra Attack and attack advantage. Your first attack hits, but doesn't crit. Should you apply Sneak Attack, or save it for the second attack which may crit but might also miss? At what value of attack roll bonus versus enemy AC does waiting become statistically advantageous? How does this change if you have the Champion's Improved Critical?
Wait, if you can hit on a roll of 7 or better
If you did hit and now take the damage immediately, you are guaranteed a 100% of your Sneak Attack damage.
The expected damage if you wait is your normal Sneak Attack damage of 100% times the likelihood to hit plus another 100% times the likelihood to crit - and if both together get you over 100%, it will be better to wait. (Since both apply to one full measure of Sneak Attack damage, I will leave out repeating "times your Sneak Attack damage" below, you can imagine that to be there).
The chance to crit for a normal critical with advantage is 9.75%. The chance for Improved Critical (on 19-20) is 19%, and the chance with Superior Critical (on 18-20) is 27.75%. So in each case, that is the amount of Sneak Attack damage that would be added to the normal Sneak Attack damage.
For normal crit chances, if you hit on a roll of 7 or better, it is better to wait. For Improved critical, you can go up to needing a roll of 9 or better, and with Superior Critical, up to needing a roll of 11 or better.
Below is a table that shows you when it is worthwile. "Roll" gives you the roll you need to roll on d20 to hit their AC with your to hit bonus, for example if it says 10, and you have +5 to hit, you would hit an AC 15 opponent; a 1 always misses, so there always remains a small chance to miss in case you roll two 1s. (There are cases where you would not even need to roll a 1 to hit, but since that is the lowest you can physically roll, the table starts there.)
"Advantage" is your likelihood to hit with advantage (and hence, expected normal Sneak Attack damage). Obviously, this is always lower than 100%. "With ___ Crit" is the relative damage you can expect to get when adding the expected damage from a crit. If this total is higher than 100%, it is worthwile to wait, because you can expect to make more damage from your next attack on average than from taking the Sneak Attack damage now.
|Roll||Advantage||With Normal Crit||With Improved Crit||With Superior Crit|
There may be other factors than just the expected damage to consider, depending on the situation. For example, if you expect a non-crit Sneak Attack to kill the target, go for it and don't wait. Conversely, if you're pretty sure a non-crit Sneak Attack won't kill but a normal hit plus a crit Sneak Attack would, you may want to wait even in cases where the analysis above suggests you shouldn't to maximize your chance of taking out the target now instead of giving them one more turn to act (thanks go to @Ilmari Karonen for pointing this out).
You also might want a bit of more saftey margin than just a fraction of a percentage point above 100%, because of unforeseen things that could get in the way before you get to make additional attacks, like an opponent having a readied action when you attack. If you are worried about that has more margin for error from the table. (Thanks to @Yakk)
Addendum: As @Ruse pointed out, you may have more than one additional attack to make, especially at higher levels. For example, if you are a level 15 fighter Champion with access to Superior Critical, you would be able to make 2 extra attacks. With Action Surge, a bonus attack from some other feature, and Haste, you could conceivably attack eight times in a round, for up to seven extra attacks. (The best use for Haste however may be to ready an Attack for their turn, so you get to Sneak Attack twice, not only once per round -- thanks to @Yakk for this finesse tip).
Having more attacks both increases you chance to hit with any one of them, and your chance to land a critical. Using the same approach as above, below are the results for having more additional attacks. This does not model the dependency of needing to both hit and crit with the same attack, so these numbers are somewhat too high, but at least they can give you a ballpark idea.
It does not list the odds for needing a 20 to hit, as if you rolled a natural 20 to hit them, you have already critted, no need to wait. (Thanks to @Yakk again.)
You can use this table when looking at each individual attack in the series that hit but did not crit, if it is worthwhile to continue trying, until you finally again arrive at the table above, when you are down to a single extra attack. The yellow highlighte cells show you what the number you need to be able to hit with is where it is still better to keep trying.
A quick trick is to "count pips" of sneak attack chances remaining.
"Pips" refer to the number of sides of a die (here a d20) that generate the result you want. We then weigh them based on crits (which count as an additional pip).
If you add up to more than 20 pips, then that is more than a full set of the result we want (delivery of sneak attack damage).
If you don't have advantage, count how many pips of hit and how many pips of crit you have on all future attacks. I.e., if you hit on a 12+, that is 9 pips of hit per attack. If you crit on a 19+, that is 2 pips of crit per attack. This sums to 11 pips of sneak attack dice per attack.
Add pips from your next attack, plus (miss chance) x pips for the attack after that (if there are 2 more), and then (miss times miss) x pips if there is yet another attack.
If that passes 20, it is worth delaying your sneak attack until a later one.
For advantage, to a pretty good approximation, your crit pips are just doubled. (20+ gives 10% rounded, 19+ gives 19% instead of 20%, 18+ gives 28% instead of 30%). For the hit pips, adding max of 50% of your pips or 50% of your miss pips is a really good approximation.
So if you hit with 8 pips, with advantage it becomes 12 pips. (40% hit chance becomes 64%, which is close to this formulas 60%; the error here is larger when the answer is going to be NO anyhow, so it works).
Hit on a 13+, crit on a 19+, 3 attacks, advantage.
Hit pips: 8 x 1.5 = 12, Miss pips = 8 (20-12), Crit pips = 4 (2 x 2).
Damage pips = 16.
With 1 more attack, not worth it. With 2, we get our 1.4x from our 40% miss rate and sum to 22 damage pips; more than 20, worth it.
For super-advantage, add 0.8x hit pips or 0.8x miss pips (whichever is smaller), and 18+ crit pips goes to 8 not 9.
Hit on a 10+, crit on an 18+, 2 attacks, super advantage.
Hit pips: 18, crit pips: 8, miss pips: 2
Total pips 26: easy peasy, save the attack.
The idea is to give something you can do mostly in your head, or at least without a calculator, at the table, and not require a table to do lookup.