Rule of thumb · MathematicsNº 104 / 156

Advantage is worth most at even odds

Rolling twice and keeping the higher result is not worth a flat bonus — it is worth nothing when you were certain either way, and a full 25 percentage points when the outcome was a coin flip.

Why it works

Keeping the best of two turns your chance of failing, p, into p squared. Squaring a number close to 0 or close to 1 barely moves it; squaring a half moves it to a quarter, which is the largest gap the operation can produce.

When it fails

It stops being true the moment the two rolls are not independent — rolling the same loaded die twice, or a second attempt that inherits the conditions that spoiled the first, does not square anything. And it says nothing about the size of a result: on a d20 advantage moves the average from 10.5 to 13.825, but if you need a natural 20 it only takes you from 5% to 9.75%.

Do it exactly

Estimate with the rule, then check it against the calculator that models it properly.

Open Dice Roller & Coin Toss →

More in Mathematics

← PreviousArea scales with the square, volume with the cube Next →A long run of heads is not evidence of anything

How much is rolling twice and keeping the best worth?

Rolling twice and keeping the higher result is not worth a flat bonus — it is worth nothing when you were certain either way, and a full 25 percentage points when the outcome was a coin flip. Keeping the best of two turns your chance of failing, p, into p squared.

Browse all 156 rules of thumb →