To calculate the probabilities of these events, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event. In the case of rolling two six-sided dice, we have 36 possible outcomes (since each die has 6 sides).
1. The number on the red die is greater than the number on the green die:
We can find the number of favorable outcomes by counting the pairs of numbers where the red die is greater than the green die. These pairs are: (2,1), (3,1), (4,1), (5,1), (6,1), (3,2), (4,2), (5,2), (6,2), (4,3), (5,3), (6,3), (5,4), (6,4), and (6,5). This gives us a total of 15 favorable outcomes. Therefore, the probability is 15/36, which simplifies to 5/12.
2. The sum is less than 10:
We can find the number of favorable outcomes by counting the pairs of numbers that have a sum less than 10. These pairs are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (5,1), (5,2), and (6,1). This gives us a total of 21 favorable outcomes. Therefore, the probability is 21/36, which simplifies to 7/12.
3. The sum is even:
To find the number of favorable outcomes, we need to count the pairs of numbers that have an even sum. These include pairs where both numbers are even (1 possible outcome: (2,2)) and pairs where one number is even and the other is odd (15 possible outcomes: (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (4,1), (4,3), (5,2), (5,4), (6,1), (6,3), (6,5)). This gives us a total of 16 favorable outcomes. Therefore, the probability is 16/36, which simplifies to 4/9.
4. The sum is prime:
The prime numbers less than or equal to 12 (the maximum sum of two dice) are 2, 3, 5, 7, and 11. We need to count the pairs of numbers that have a sum equal to one of these primes: (1,1), (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (4,1), (4,3), (5,2), (5,6), (6,1), (6,5). This gives us a total of 15 favorable outcomes. Therefore, the probability is 15/36, which simplifies to 5/12.
5. The sum is 7 or 11:
We need to count the pairs of numbers that have a sum equal to 7 or 11. These pairs are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). This gives us a total of 6 favorable outcomes. Therefore, the probability is 6/36, which simplifies to 1/6.