Explanation:

When two dice are thrown simultaneously, there are 36 possible outcomes. The sum of the two numbers can range from 2 to 12.

Finding the probability of the sum being divisible by 3:

If the sum is divisible by 3, then the possible sums are 3, 6, 9, and 12.

To find the probability of the sum being divisible by 3, we need to count the number of outcomes that give us those sums.

- For a sum of 3, there is only one possible outcome: (1,2).
- For a sum of 6, there are four possible outcomes: (1,5), (2,4), (3,3), and (4,2).
- For a sum of 9, there are four possible outcomes: (3,6), (4,5), (5,4), and (6,3).
- For a sum of 12, there is only one possible outcome: (6,6).

Therefore, there are 10 possible outcomes that give us a sum divisible by 3. So the probability of the sum being divisible by 3 is 10/36 or 5/18.

Finding the probability of the sum being divisible by 4:

If the sum is divisible by 4, then the possible sums are 4, 8, and 12.

- For a sum of 4, there is only one possible outcome: (1,3).
- For a sum of 8, there are five possible outcomes: (2,6), (3,5), (4,4), (5,3), and (6,2).
- For a sum of 12, there is only one possible outcome: (6,6).

Therefore, there are 7 possible outcomes that give us a sum divisible by 4. So the probability of the sum being divisible by 4 is 7/36.

Finding the probability of the sum being divisible by 3 or 4:

To find the probability of the sum being divisible by 3 or 4, we need to add the probabilities of the sum being divisible by 3 and the sum being divisible by 4. However, we need to subtract the probability of the sum being divisible by both 3 and 4 (i.e., 12). This is because we don't want to count the outcome (6,6) twice.

So the probability of the sum being divisible by 3 or 4 is:

5/18 + 7/36 - 1/36 = 17/36

Therefore, the probability of the sum being divisible by 3 or 4 is 17/36.