Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is divisible by 4 or 6?

To solve this problem, we first need to understand the total number of outcomes when two dice are thrown together. Since a die has 6 faces, the total number of outcomes is 6*6 = 36.

Next, we need to find out the favorable outcomes i.e., the outcomes where the sum of the numbers on the two faces is divisible by 4 or 6.

The sums that are divisible by 4 are 4 and 8. The pairs that give these sums are (1,3), (3,1), (2,2), (1,7), (7,1), (2,6), (6,2), (3,5), (5,3), (4,4). So, there are 10 such pairs.

The sums that are divisible by 6 are 6 and 12. The pairs that give these sums are (1,5), (5,1), (2,4), (4,2), (3,3), (1,11), (11,1), (2,10), (10,2), (3,9), (9,3), (4,8), (8,4), (5,7), (7,5), (6,6). So, there are 16 such pairs.

But we have double counted pairs that give sums divisible by both 4 and 6 i.e., (2,2), (4,4), (1,5), (5,1), (2,4), (4,2), (3,3). There are 7 such pairs.

So, the total number of favorable outcomes is 10 + 16 - 7 = 19.

Therefore, the probability that the sum of the numbers on the two faces is divisible by 4 or 6 is 19/36.