The Mathematics of Advantage: When to Take the Risk - ICE5e Blog

One of 5th Edition's most elegant design choices is the Advantage/Disadvantage system. Instead of tracking floating modifiers like "+2 for flanking, -4 for cover, +1 for high ground" (common in 3.5e and Pathfinder), you simply roll two d20s and take the highest (or lowest).

It feels intuitive and keeps the game moving, but the actual mathematics behind it are often misunderstood. Unlike a flat +X bonus, the value of Advantage is dynamic—it changes based on how difficult the task is. Knowing this probability curve can help you make better tactical decisions.

The "Plus Five" Myth

A common rule of thumb is that Advantage is worth a +5 bonus. This is derived from the passive check rules (Passive Perception increases by +5 with advantage).

However, mathematically, the value changes depending on the target number (DC or AC) you are trying to hit. Advantage is most valuable when you need a "middle of the road" roll.

Target Roll Needed	Win Chance (Normal)	Win Chance (Advantage)	Effective Bonus
20 (Natural 20)	5%	9.75%	~ +1
15	30%	51%	~ +4
11	50%	75%	+5
8	65%	87.75%	~ +4.5
2	95%	99.75%	~ +1

Key Takeaway: Advantage is most valuable when you need to roll an 8 to 14 on the die. If the DC is extremely high (you need a 19 or 20) or extremely low, Advantage helps less than a flat numerical bonus (like Bless or a +1 weapon) would.

Great Weapon Master & Sharpshooter

These feats allow you to take a -5 penalty to hit for +10 damage. This is a massive damage boost, but the accuracy penalty is steep. When does the math say you should use it?

The math suggests you should toggle "Power Attack" on when:

You have Advantage: This is the biggest factor. Advantage raises your average roll from 10.5 to 13.8. This statistical buffer effectively negates the -5 penalty.

The Target AC is Low: If you only need a 5 to hit (e.g., attacking a Zombie or Ooze), the -5 penalty still leaves you needing a 10 (55% chance). The extra damage is worth the risk.

Your Damage Die is Small:

- Hand Crossbow (1d6 + 3): Avg 6.5 dmg. Adding +10 is a 150% increase. Huge value. - Paladin Smite (2d6 + 4d8 + 4): Avg 30 dmg. Adding +10 is only a 33% increase. The risk of missing that big smite might not be worth the +10 flat damage.

ICE5e's attack calculator handles the to-hit math for you, but deciding when to toggle that GWM switch is a tactical skill.

The "Elven Accuracy" Curve

The Elven Accuracy feat allows you to reroll one of the dice when you have Advantage on Dex, Int, Wis, or Cha attacks. This is often called "Triple Advantage."

While normal Advantage increases your average roll to 13.82, Elven Accuracy pushes it to 15.49. More importantly, it drastically increases your Critical Hit chance:

Normal: 5%
Advantage: 9.75%
Elven Accuracy: 14.26%

This makes Elven Accuracy a top-tier pick for "Crit Fishing" builds, like Paladins (Divine Smite) or Rogues (Sneak Attack), where doubling the damage dice is impactful.

Disadvantage is Deadly

We often focus on getting Advantage, but we underestimate Disadvantage. Mathematically, Disadvantage is devastating against high DCs.

If you need a 15 to hit, your chance drops from 30% to 9%.
If you need a 18, your chance drops from 15% to 2.25%.

This is why spells like Blur, Cloak of Displacement, or the Dodge action are so powerful for high-AC characters. If you have AC 20 and the enemy needs a 15 to hit you, imposing Disadvantage makes you nearly invincible (91% chance they miss).

The "Lucky" Feat Weirdness

The Lucky feat lets you roll an extra d20.

Normal Roll: You turn a single roll into "Super Advantage" (pick best of 2).
With Disadvantage: This is where it gets weird. By RAW (Rules as Written), Lucky lets you roll a 3rd die and pick any of the 3.

- Example: You have Disadvantage. You roll a 2 and a 18. You must take the 2. - You use Lucky. You roll a 15. - You now choose the best of {2, 18, 15}. You take the 18.

This turns Disadvantage into "Super Advantage." (Note: Many DMs house-rule this to simply cancel out Disadvantage, so check with your table!).

Summary

Advantage gives roughly a +3.325 increase to your average roll (from 10.5 to 13.825).
It significantly reduces the chance of a Critical Miss (from 5% to 0.25%).
It effectively cancels out the penalty of Great Weapon Master / Sharpshooter.

Understanding these odds helps you decide when to burn resources (like Inspiration, Faerie Fire, or Reckless Attack) to gain that tactical edge.