Question

three dice are rolled and told that exactly two of them are showing the same number the probability of getting sum 16

Answer 1

Given that exactly two of the three dice show the same number, there are six possibilities for the number shown on two of the dice, and five possibilities for the number shown on the other die. Therefore, there are 6*5 = 30 equally likely possible outcomes to consider:

1-1-2, 1-1-3, 1-1-4, 1-1-5, 1-1-6

2-2-1, 2-2-3, 2-2-4, 2-2-5, 2-2-6

3-3-1, 3-3-2, 3-3-4, 3-3-5, 3-3-6

4-4-1, 4-4-2, 4-4-3, 4-4-5, 4-4-6

5-5-1, 5-5-2, 5-5-3, 5-5-4, 5-5-6

6-6-1, 6-6-2, 6-6-3, 6-6-4, 6-6-5

Of these 30 possible outcomes, only these two: 556 and 664, produce a sum of 16 on the three dice combined.

Therefore, the probability of a sum of 16 given exactly two of the three dice match, is 2/30 = 1/15