The most likely total of three dice

Question

Two fair dice are most likely to total 7, with probability 6/36.

With three fair dice, which total (or totals) is most likely, and what is the probability? Also find the probability of rolling a total of exactly 9, which was famously argued about in the 17th century.

Accepted Answer

Totals of 10 and 11 are the most likely, each with probability 27/216 = 1/8. A total of 9 has probability 25/216. Symmetry. Replacing each die's value v by 7 minus v maps a total t to 21 minus t, so the distribution is symmetric about 10.5. The peak must therefore be at 10 and 11 jointly. Counting 10. List the unordered combinations of three dice adding to 10 and count how many orderings each has: - 1, 3, 6: 6 orderings. 1, 4, 5: 6. 2, 2, 6: 3. 2, 3, 5: 6. 2, 4, 4: 3. 3, 3, 4: 3. Total: 6 + 6 + 3 + 6 + 3 + 3 = 27. By symmetry, 11 also has 27. Each has probability 27/216 = 1/8. Counting 9. 1, …

The most likely total of three dice

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