I roll a d20.

If I have a choice between d20+x with advantage, or d20+y without advantage, what choice do I make to maximise the result? What are the values of x and y where the choice changes (if it changes at all)?

Assume x < y.

Theory

The first thing we'll look at is a table that represents the odds of rolling at least a given DC, given a d20 with or without advantage (no modifiers yet).

begin{array}{r|llll}
text{Natural DC} & text{Advantage} & text{No Advantage} & text{Difference} & text{Eq. Flat Modifier} \ hline
text{DC 1-} & text{100.000%} & text{100.000%} & text{0.000%} & 0 \
text{DC 2} & text{99.750%} & text{95.000%} & text{4.750%} & 0.95 (1)\
text{DC 3} & text{99.000%} & text{90.000%} & text{9.000%} & 1.8 (2) \
text{DC 4} & text{97.750%} & text{85.000%} & text{12.750%} & 2.55 (3) \
text{DC 5} & text{96.000%} & text{80.000%} & text{16.000%} & 3.2 (4) \
text{DC 6} & text{93.750%} & text{75.000%} & text{18.750%} & 3.75 (4) \
text{DC 7} & text{91.000%} & text{70.000%} & text{21.000%} & 4.2 (5) \
text{DC 8} & text{87.750%} & text{65.000%} & text{22.750%} & 4.55 (5) \
text{DC 9} & text{84.000%} & text{60.000%} & text{24.000%} & 4.8 (5) \
text{DC 10} & text{79.750%} & text{55.000%} & text{24.750%} & 4.95 (5) \
text{DC 11} & text{75.000%} & text{50.000%} & text{25.000%} & 5 \
text{DC 12} & text{69.750%} & text{45.000%} & text{24.750%} & 4.95 (5) \
text{DC 13} & text{64.000%} & text{40.000%} & text{24.000%} & 4.8 (5) \
text{DC 14} & text{57.750%} & text{35.000%} & text{22.750%} & 4.55 (5) \
text{DC 15} & text{51.000%} & text{30.000%} & text{21.000%} & 4.2 (5) \
text{DC 16} & text{43.750%} & text{25.000%} & text{18.750%} & 3.75 (4) \
text{DC 17} & text{36.000%} & text{20.000%} & text{16.000%} & 3.2 (4) \
text{DC 18} & text{27.750%} & text{15.000%} & text{12.750%} & 2.55 (3) \
text{DC 19} & text{19.000%} & text{10.000%} & text{9.000%} & 1.8 (2) \
text{DC 20} & text{9.750%} & text{5.000%} & text{4.750%} & 0.95 (1) \
text{DC 21+} & text{0.000%} & text{0.000%} & text{0.000%} & 0 \
end{array}

A +1 to a non-advantage roll will always improve the odds of rolling a given number by exactly 5 percentage points. Conversely, a +1 to an Advantage roll will increase your odds by an amount equal to moving up one row on that table: a DC7 check made with +1 is equivalent to a DC6 check made with +0. A DC20 check made with advantage and a +1 modifier is equivalent to a DC19 check made with +0, which constitutes a 9.250 percentage point improvement.

There are a few casual observations we can make:

• It's not possible to roll a natural d20 lower than a 1, so if if the advantage check requires a natural 1, then there's no benefit to gaining any modifier (or advantage, for that matter): it's a check that is impossible to fail.


• At DC 2, gaining advantage increases the odds of success by 4.750% (to 99.750%) but gaining a +1 modifier increases the odds of success by 5% (to 100%). So intuitively, If we're comparing 1d20+x/ADV vs 1d20+x+1/NoADV, and the natural number we need to hit is a 2 (for the advantage check), then the +1 modifier is better.


• It's the same deal at DC20: Gaining Advantage will improve from 5% to 9.750%, but gaining +1 will improve from 5% to 10%. Again, the +1 modifier is better.


• But the differences get more drastic as we move closer to the mean of the roll: At DC3, advantage improves the odds by 9% (90%→99%) but a +1 modifier only improves the odds by 5% (90%→95%), so here, Advantage is better than a +1 modifier; but it's NOT better than a +2 modifier (90%→100%).


• In the table, I've added the "Eq. Flat Modifier" column: this describes, for each row, how much of a modifier you would need for the benefit from that modifier to be equivalent to the benefit provided by Advantage. Since 5e doesn't have "half" modifiers or fractional DCs, I've included the (rounded up) proper modifier in parenthesis next to it. In each row, if the modifier difference between the Advantage and Non-Advantage rolls is greater than that number, then the modifier is better; if it's not, then the Advantage roll is better.

Practice

So back to the original question: Given two rolls, 1d20+x/Adv, and 1d20+y/NoAdv, which is better? Well, as established, it depends on the DC of the check, but to get the results from this table:

• Calculate the difference between y and x


• Subtract x (the modifier for the Advantage roll) from the DC to get the "Natural DC"


• Look at the Eq. Flat Modifier for that row in the table


• If the difference between y and x is greater than that value, then you should prefer the 1d20+y/NoAdv roll. If not, then you should prefer the 1d20+x/Adv roll.

Examples

• DC19, 1d20+5/Adv vs 1d20+7/NoAdv
  
  
  
  
  • Difference is 7 - 5 == 2
  
  
  • DC19 - 5 is DC14
  
  
  • DC14 has a Eq. Flat Modifier of 4.55
  
  
  • Therefore, the Advantage roll is better than the non-Advantage roll.
  
  
  
  


• DC3, 1d20+1/Adv vs 1d20+2/NoAdv
  
  
  
  
  • Difference is 2 - 1 == 1
  
  
  • DC3 - 1 is DC2
  
  
  • DC2 has a Eq. Flat Modifier of 0.95
  
  
  • Therefore, the non-advantage roll is better than the Advantage roll
  
  
  
  


• DC17, 1d20+9/Adv vs 1d20+14/NoAdv
  
  
  
  
  • We could skip the steps: none of the rows have a Eq. Flat Modifier greater than 5, meaning the +5 modifier will always be better than (or equivalent to) the Advantage improvement. Nonetheless...
  
  
  • Difference is 14 - 9 == 5
  
  
  • DC17 - 9 is DC8
  
  
  • DC8 has a Eq. Flat Modifier of 4.55
  
  
  • Therefore, the non-advantage roll is better than the Advantage roll
  
  
  
  

Attack Rolls

Attack Rolls are a little weird, because you no longer simply care about passing the check; you also care what the natural number was because of Critical Hits and Misses.

Most of the math still checks out: if all you care about is hitting/missing, then the table above can be used, since the scenarios where a Natural 2 hits and a Natural 19 misses are pretty rare in 5e. If, however, you instead care more about the Crits/Auto-Misses, then you should introduce a "subjectivity factor", which you can define however you like: is it important to you that you get a critical hit (or avoid a critical miss)? Then always go Advantage. If not, then use the table above. I generally stick to the table personally, but "clutch factor" is one of those hazy things that can't be objectively defined, so you'll need to make that call for yourself.

It will depend on what you're trying to achieve. For example, if you need to reach DC 25, and $x = 4$ and $y = 5$, the advantage on the roll with $x$ doesn't matter; you'll never roll higher than 24. With the +5, you'll at least have a 5% chance.

Here (scroll down to "Advantage versus Simple Bonuses") is a table which shows which bonus (difference between x and y) corresponds to having advantage or not.

(source: Zero Hit Points)

If you only care about maximising the expected result, as opposed to your odds of hitting a specific target number (for instance, you might be making a contested roll against someone else, like in a grapple, or otherwise don't know the target number ahead of time) this is a pretty simple comparison. Having advantage on a d20 roll increases the expected result from an average of 10.5 to 13.82 (illustrated by this anydice program); that's a benefit of +3.32.

Therefore, in order for a roll without advantage to have a higher expected result than a roll with advantage, the modifier on the normal roll needs to be four or more points better than the modifier on the advantaged roll. +3 with advantage is worse than +7 normally, and so on.

First, subtract $x$ both from $y$ and from the target number you're rolling against. Then look at this graph:

In the graph, find the position on the horizontal axis that matches the target number (minus $x$) that you're trying to meet or exceed, and the colored line that matches the extra bonus $y-x$ to the roll without advantage. If that colored line is higher that the curved black line at that position on the horizontal axis, you should choose the higher bonus over advantage.

(Specifically, the various lines in the graph show the probability of meeting or exceeding a given target number with various rolls: the black curved line is for d20 with advantage but no bonus, while the five different colored straight lines on top of it are for d20+1 to d20+5.)

Or, to summarize, you should choose a plain $+y$ bonus over advantage $+x$ when...

• 
  $y = x + 1$ and the target number is at most $x+2$ or at least $x+20$;


• 
  $y = x + 2$ and the target number is at most $x+3$ or at least $x+19$;


• 
  $y = x + 3$ and the target number is at most $x+4$ or at least $x+18$;


• 
  $y = x + 4$ and the target number is at most $x+6$ or at least $x+16$; or


• 
  $y ge x + 5$.

(As noted by Xirema, things can change a little if you're e.g. making an attack roll and care about crits. Rolling with advantage has a 9.75% chance of giving you a natural 20, and only a 0.25% chance of a natural 1, whereas with a normal d20 roll both 1s and 20s show up 5% of the time each. Whether those differences in crit odds are worth trading for a somewhat worse chance to hit depends both on the target DC and on how much you value crits.)