D&D Advantage Probability: Is It Really a +5?

Advantage in D&D 5e means rolling two d20s and keeping the higher, so your chance of meeting a target number X becomes 1 − ((X−1)/20)². It peaks at the equivalent of a +5 bonus when you need an 11, stays near +4 to +5 across the common DC 8 to 14 band, but averages closer to +3 overall and shrinks to about +1 at the extremes.

What advantage actually does to the dice

A plain d20 is flat: every face from 1 to 20 has the same 5% chance. Advantage changes the shape of the distribution rather than sliding it sideways. When you roll twice and keep the higher die, large numbers become far more likely and small numbers become rare. The probability of rolling exactly the value x with advantage works out to a clean line, (2x − 1)/400. So a 1 appears only 1 in 400 times (0.25%), while a 20 appears 39 in 400 times (about 9.75%) instead of the usual 5%.

That reshaping is why the average roll jumps. A normal d20 averages 10.5. With advantage the expected value climbs to 13.825, and with disadvantage (keep the lower) it falls to 7.175. The gap between a normal roll and advantage is therefore about 3.3 in pure expected value, which is the first hint that the famous "+5" is not the whole story.

The core formula: 2p minus p squared

For most rolls you do not care about the exact face, only whether you hit a target number or higher. To succeed with advantage, you need at least one of the two dice to land on target. It is easier to compute the chance of failing, which requires both dice to miss. If a single die succeeds with probability p, then both fail with probability (1 − p)², so the advantage success chance is:

P(success with advantage) = 1 − (1 − p)² = 2p − p²

That parabola, 2p − p², is the engine behind every advantage claim. Plug in the d20 directly: needing an 11 or better is a single-die p of 0.5, so advantage gives 1 − (10/20)² = 1 − 0.25 = 0.75. Your odds leap from 50% to 75%. Disadvantage flips the sign and uses p², dropping that same 50% roll to 25%.

Where the +5 comes from and where it breaks

To turn advantage into an "effective bonus," ask what flat modifier on a single d20 would give the same success chance. Needing an 11 with advantage yields 75% success. A lone d20 also hits 75% when you only need a 6. Going from needing 11 down to needing 6 is a 5-point improvement, so advantage is worth exactly +5 right at the 50/50 point. This is the single best case, the peak of the curve, not the average.

The reason +5 feels right at the table is that the boost stays generous across the whole playable middle. If the number you need is anywhere from 8 to 14, the best-matching flat bonus is +4 or +5 for every one of those targets. Since a huge share of real rolls in play, attacks against typical armor classes and skill checks against DC 10 to 15, land in that band, advantage genuinely behaves like a +5 most of the time you actually roll. The calculator at /calculators/dnd-advantage turns any target into its exact normal, advantage, and disadvantage odds so you can see the band yourself.

The curve collapses at the edges. If you need a 20, a single die succeeds 5% of the time, and advantage gives 1 − (19/20)² = 0.0975, just 9.75%. That is the same as needing roughly a 19 on one die, an improvement of only about +1. At the low end the bonus also shrinks, because if you only need a 2 you were going to succeed 95% of the time anyway and advantage has almost no room to help. Averaged across all 20 possible targets the effective bonus settles near +3, well below the headline number, which is why some tables quote "+3" or "+4" as the honest all-around figure.

Why the rules just say add 5

Advantage is the signature mechanic of 5th edition, introduced in the 2012 "D&D Next" playtest and kept because it folds dozens of fiddly situational modifiers from earlier editions into one elegant rule. But there is one place where you cannot roll twice: passive checks. A passive Perception score has no dice at all, it is a fixed number representing baseline awareness. The Player's Handbook (page 175) handles advantage on a passive check with a flat instruction, add 5, and for disadvantage subtract 5.

That printed +5 is the designers picking the peak-of-the-curve value as a simple stand-in, because a passive score sits right in the middle of the action where advantage really is worth about +5. A wolf's Keen Hearing and Smell, for instance, lifts its passive Perception from 13 to 18 for scent and sound. So the "+5" is simultaneously a rules-as-written constant for passive checks and a rough average for active rolls, which is exactly why the number is both everywhere and slightly misleading.

Practical takeaways at the table

• Advantage shines on coin-flip rolls. The closer your odds are to 50%, the more advantage helps, peaking at a full +5 when you need an 11.
• Advantage cannot break the ceiling. You can never exceed 20 on the die, so against a target needing a natural 20 you only gain about +1. A real +4 or +5 modifier would still be better there because it can push a total past 20.
• It barely matters when you are already likely to succeed. Stacking advantage on a roll you make 90% of the time is nearly wasted; save the source for a tense check.
• Critical hits love advantage. Two d20s nearly double your chance of a natural 20, from 5% to 9.75%, which matters for crit-fishing builds even when the to-hit boost is small.

The deeper lesson is that advantage is a probability-reshaping tool, not a flat bonus, and the same "keep the better of two tries" logic shows up far beyond the gaming table. If you enjoy seeing how a simple rule bends a distribution, the broader /calculators/probability tools and the /calculators/standard-deviation calculator let you explore how spread and skew behave once you stop assuming everything is a clean flat line.

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