Odds and tables
Dice odds: every total, worked out
Why 7 keeps coming up, what “advantage” is really worth, and the exact odds for any handful of dice you care to throw.
Skip to the calculatorOne die is simple. Two dice have opinions.
A single fair die has no favourites: each face comes up one time in six, and that is the whole story. Add a second die and sum the faces, and suddenly the middle totals crowd out the ends. There is only one way to roll a 2 (1 and 1) but six ways to roll a 7 — 1+6, 2+5, 3+4 and their mirror images. That is why 7 lands one roll in six, and 2 or 12 only one roll in thirty-six.
| Total | Ways to make it | Chance | Share |
|---|---|---|---|
| 2 | 11+1 | 1/36 · 2.8% | |
| 3 | 21+2, 2+1 | 2/36 · 5.6% | |
| 4 | 31+3, 2+2, 3+1 | 3/36 · 8.3% | |
| 5 | 41+4, 2+3, 3+2, 4+1 | 4/36 · 11.1% | |
| 6 | 51+5, 2+4, 3+3, 4+2, 5+1 | 5/36 · 13.9% | |
| 7 | 61+6, 2+5, 3+4, 4+3, 5+2, 6+1 | 6/36 · 16.7% | |
| 8 | 52+6, 3+5, 4+4, 5+3, 6+2 | 5/36 · 13.9% | |
| 9 | 43+6, 4+5, 5+4, 6+3 | 4/36 · 11.1% | |
| 10 | 34+6, 5+5, 6+4 | 3/36 · 8.3% | |
| 11 | 25+6, 6+5 | 2/36 · 5.6% | |
| 12 | 16+6 | 1/36 · 2.8% |
Three dice, and the puzzle that fooled gamblers.
With three dice the totals 9 and 10 each have six “unordered” recipes (for 9: 1+2+6, 1+3+5, 1+4+4, 2+2+5, 2+3+4, 3+3+3), so it is tempting to call them equally likely. They are not. Count the dice as distinct — a red, a green and a blue — and 10 can be made in 27 ways to 9’s 25, because a recipe like 2+3+4 can land in six arrangements while 3+3+3 can land in only one. Over 216 rolls, 10 shows up about two more times than 9. Gamblers noticed the difference long before anyone could explain it.
Advantage, in numbers.
Tabletop games often let you roll a d20 twice and keep the better result (“advantage”) or the worse one (“disadvantage”). The effect is biggest in the middle of the range: a target you hit half the time on one die becomes a 75% shot with advantage and a 25% shot with disadvantage. At the extremes it matters less in absolute terms — a natural 20 goes from 5% to 9.75%.
| Need at least | One die | Advantage | Disadvantage |
|---|---|---|---|
| 20 | 5.0% | 9.8% | 0.3% |
| 19 | 10.0% | 19.0% | 1.0% |
| 18 | 15.0% | 27.8% | 2.3% |
| 17 | 20.0% | 36.0% | 4.0% |
| 16 | 25.0% | 43.8% | 6.3% |
| 15 | 30.0% | 51.0% | 9.0% |
| 14 | 35.0% | 57.7% | 12.2% |
| 13 | 40.0% | 64.0% | 16.0% |
| 12 | 45.0% | 69.8% | 20.3% |
| 11 | 50.0% | 75.0% | 25.0% |
| 10 | 55.0% | 79.8% | 30.3% |
| 8 | 65.0% | 87.8% | 42.3% |
| 6 | 75.0% | 93.8% | 56.3% |
| 4 | 85.0% | 97.8% | 72.2% |
| 2 | 95.0% | 99.8% | 90.3% |
Four quick answers.
- At least one six in four rolls of a die? 1 − (5/6)4 = 51.8%. Slightly better than even.
- At least one double six in 24 rolls of two dice? 1 − (35/36)24 = 49.1%. Slightly worse than even — a famous pair of bets that look alike and aren’t.
- A natural 20 at some point in 20 rolls? 1 − 0.9520 = 64.2%.
- Rolling the same face on five dice (“Yahtzee”) in one throw? 6/65 = 1 in 1,296, or 0.08%.
Work out the odds for your handful.
Choose how many dice and how many sides. The chart shows the exact chance of every total; then roll a thousand times and see how reality hugs the curve.
Exact chance of each total
What 1,000 rolls actually gave
The chart shows the exact chance of every total.
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