A once very difficult but now very simple problem in probability from Ian Stewart’s Do Dice Play God? (2019):
For three dice [Girolamo] Cardano solved a long-standing conundrum [in the sixteenth century]. Gamblers had long known from experience that when throwing three dice, a total of 10 is more likely than 9. This puzzled them, however, because there are six ways to get a total of 10:
1+4+5; 1+3+6; 2+4+4; 2+2+6; 2+3+5; 3+3+4
But also six ways to get a total of 9:
1+2+6; 1+3+5; 1+4+4; 2+2+5; 2+3+4; 3+3+3
So why does 10 occur more often?
To see the answer, imagine throwing three dice of different colors: red, blue and yellow. How many ways can you get 9 and how many ways can you get 10?
| Roll | Total=9 | Dice #1 (Red) | Dice #2 (Blue) | Dice #3 (Yellow) |
| 01 | 9 = | 1 | 2 | 6 |
| 02 | 9 = | 1 | 3 | 5 |
| 03 | 9 = | 1 | 4 | 4 |
| 04 | 9 = | 1 | 5 | 3 |
| 05 | 9 = | 1 | 6 | 2 |
| 06 | 9 = | 2 | 1 | 6 |
| 07 | 9 = | 2 | 2 | 5 |
| 08 | 9 = | 2 | 3 | 4 |
| 09 | 9 = | 2 | 4 | 3 |
| 10 | 9 = | 2 | 5 | 2 |
| 11 | 9 = | 2 | 6 | 1 |
| 12 | 9 = | 3 | 1 | 5 |
| 13 | 9 = | 3 | 2 | 4 |
| 14 | 9 = | 3 | 3 | 3 |
| 15 | 9 = | 3 | 4 | 2 |
| 16 | 9 = | 3 | 5 | 1 |
| 17 | 9 = | 4 | 1 | 4 |
| 18 | 9 = | 4 | 2 | 3 |
| 19 | 9 = | 4 | 3 | 2 |
| 20 | 9 = | 4 | 4 | 1 |
| 21 | 9 = | 5 | 1 | 3 |
| 22 | 9 = | 5 | 2 | 2 |
| 23 | 9 = | 5 | 3 | 1 |
| 24 | 9 = | 6 | 1 | 2 |
| 25 | 9 = | 6 | 2 | 1 |
| Roll | Total=10 | Dice #1 (Red) | Dice #2 (Blue) | Dice #3 (Yellow) |
| 01 | 10 = | 1 | 3 | 6 |
| 02 | 10 = | 1 | 4 | 5 |
| 03 | 10 = | 1 | 5 | 4 |
| 04 | 10 = | 1 | 6 | 3 |
| 05 | 10 = | 2 | 2 | 6 |
| 06 | 10 = | 2 | 3 | 5 |
| 07 | 10 = | 2 | 4 | 4 |
| 08 | 10 = | 2 | 5 | 3 |
| 09 | 10 = | 2 | 6 | 2 |
| 10 | 10 = | 3 | 1 | 6 |
| 11 | 10 = | 3 | 2 | 5 |
| 12 | 10 = | 3 | 3 | 4 |
| 13 | 10 = | 3 | 4 | 3 |
| 14 | 10 = | 3 | 5 | 2 |
| 15 | 10 = | 3 | 6 | 1 |
| 16 | 10 = | 4 | 1 | 5 |
| 17 | 10 = | 4 | 2 | 4 |
| 18 | 10 = | 4 | 3 | 3 |
| 19 | 10 = | 4 | 4 | 2 |
| 20 | 10 = | 4 | 5 | 1 |
| 21 | 10 = | 5 | 1 | 4 |
| 22 | 10 = | 5 | 2 | 3 |
| 23 | 10 = | 5 | 3 | 2 |
| 24 | 10 = | 5 | 4 | 1 |
| 25 | 10 = | 6 | 1 | 3 |
| 26 | 10 = | 6 | 2 | 2 |
| 27 | 10 = | 6 | 3 | 1 |
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