What is the probability of rolling a total of 9 with three six-sided dice?
Instruction: Assume three fair, independent six-sided dice. List the unordered patterns that sum to 9, count their ordered permutations, and calculate the probability.
Context: This question assesses the candidate's ability to think in multidimensional probability spaces with multiple dice.
Updated
Worked answer
Assume the three dice are fair and independent. Treat them as distinguishable, so there are 6³ = 216 equally likely ordered outcomes.
The unordered patterns summing to 9 and their numbers of permutations are:
| Pattern | Ordered outcomes |
|---|---|
| (1, 2, 6) | 6 |
| (1, 3, 5) | 6 |
| (1, 4, 4) | 3 |
| (2, 2, 5) | 3 |
| (2, 3, 4) | 6 |
| (3, 3, 3) | 1 |
The count is 6 + 6 + 3 + 3 + 6 + 1 = 25, giving 25/216 ≈ 0.115741, or 11.5741%. Counting each pattern once would be incorrect because the patterns have different multiplicities.
from itertools import product
outcomes = list(product(range(1, 7), repeat=3))
assert sum(sum(roll) == 9 for roll in outcomes) == 25
assert len(outcomes) == 216