The probability of rolling a total of 10 with three fair dice can be calculated as follows.
First, identify the total number of possible outcomes when rolling three dice:
Total outcomes
=6×6×6=216Next, find the number of successful outcomes where the sum of the dice equals 10. Consider each possible combination:
Combination: (1, 3, 6), (1, 6, 3), (3, 1, 6), etc. (permutations):
Number of outcomes:
3!=6Combination: (2, 4, 4), (4, 4, 2), (4, 2, 4) (permutations of two identical numbers):
Number of outcomes:
‌=3Combination:
(2,3,5),(3,2,5),(5,3,2), etc. (permutations):
Number of outcomes:
3!=6Combination: (3, 3, 4), (3, 4, 3), (4, 3, 3) (permutations of two identical numbers):
Number of outcomes:
‌=3Combination:
(2,2,6),(2,6,2),(6,2,2) (permutations of two identical numbers):
Number of outcomes:
‌=3Combination: (3, 5, 2), (5, 3, 2), (2, 3, 5), etc. (permutations):
Number of outcomes:
3!=6The total number of favorable outcomes for rolling a sum of 10 is:
6+3+6+3+3+6=27Finally, calculate the probability:
Probability
=‌| ‌ Favorable outcomes ‌ |
| ‌ Total outcomes ‌ |
=‌=‌