The maths of keep-highest & keep-lowest

Many of the best-loved dice mechanics involve rolling more dice than you need and throwing some away. Keep the highest and you tilt results upward; keep the lowest and you drag them down. The effect feels intuitive, but the exact size of the shift surprises most players — and it is worth knowing before you build a house rule around it.

The baseline: a plain sum

Start from the honest average. Because expected values simply add, three six-sided dice average 3 × 3.5 = 10.5, spread symmetrically around that centre. That is our reference point. The moment we roll a fourth die and drop one, the symmetry breaks.

Keep-highest raises the average

Consider 4d6, drop the lowest — the classic way to roll a D&D ability score. You roll four dice and keep the best three. Because the worst die is discarded, small numbers are far more likely to be the one thrown away, so the kept three lean high:

So the single extra die and one drop is worth roughly +1.7 to the average and pushes the peak of the curve up to 13. It also skews the shape: high scores (15–18) become much more common while very low scores grow rare — rolling a 3 now needs all four dice to come up 1, a mere 1-in-1296 chance. This is precisely why characters rolled with 4d6-drop-lowest tend to out-stat the “fair” point-buy baseline.

Advantage: keep-highest on a d20

The same principle drives advantage in D&D 5e — roll two d20 and keep the higher. A lone d20 averages 10.5; take the higher of two and the average climbs to 13.825, a raw lift of about +3.3 on the number you roll. But an average hides the real behaviour: advantage helps most when your target sits near the middle of the die and barely at all at the extremes. (We break that curve down fully in advantage vs flat bonuses.)

Keep-lowest is the exact mirror

Keep-lowest — disadvantage — is keep-highest reflected. Whatever a keep-highest rule adds above the plain single-die average, the matching keep-lowest rule subtracts by the same amount. The two always balance around the ordinary mean:

Notice how each keep-highest and keep-lowest pair sits equal distances above and below the single-die average — 13.825 and 7.175 are both 3.325 away from 10.5. That symmetry is why advantage and disadvantage feel like clean opposites: they are, mathematically, the same operation pointed in opposite directions.

It reshapes the spread, not just the mean

Keeping dice does more than move the average — it changes the shape of the outcomes. Rolling extra dice and keeping the best clusters your results toward the top end and thins out the low tail, so you not only score higher on average but also more consistently high. Keep-lowest does the reverse, bunching results low. This is why advantage feels reliable in a way a flat bonus does not: it does not just nudge the centre, it reshapes where your rolls tend to land.

Why it matters at the table

• Ability scores run hot. 4d6-drop-lowest averages about 12.24 per score versus the standard array’s more modest spread — expect stronger characters.
• Adding dice to drop is a big lever. Even one extra die and a single drop shifts the average by well over a point; stacking more amplifies it fast.
• Advantage is not a fixed bonus. Its +3.3 average understates its value on coin-flip rolls and overstates it at the extremes.

One more subtlety worth knowing as a designer or a player: the more dice you roll before dropping, the larger the shift, but with diminishing returns. Going from 3d6 to 4d6-drop-lowest adds about 1.7 to the average; adding a fifth die and dropping the two lowest lifts it further, yet each extra die you discard buys a little less than the last. If you want a dramatic swing, add one die; if you want near-guaranteed high results, you must add several.

The surest way to feel these shifts is to watch them happen. Roll 4d6-drop-lowest a few dozen times in Dice Drop and note how rarely you dip below 10 — the discarded die is doing quiet, steady work.

Keep reading: advantage vs flat bonuses — when a humble +2 quietly beats a second die.