1. **Problem statement:** Calculate the probability for each of the following when rolling two six-sided dice: - Getting the same number on each roll. - Obtaining an odd prime number (excluding 1) on each roll. - The difference of the two numbers is at most 1. - The second number is a multiple of the first number. 2. **Total possible outcomes:** Each die has 6 faces, so total outcomes for two dice are $$6 \times 6 = 36$$. 3. **Probability of getting the same number on each roll:** - Possible pairs: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) - Number of favorable outcomes: 6 - Probability: $$\frac{6}{36} = \frac{1}{6}$$ 4. **Probability of obtaining an odd prime number on each roll:** - Odd primes on a die: 3 and 5 - Possible pairs: (3,3), (3,5), (5,3), (5,5) - Number of favorable outcomes: 4 - Probability: $$\frac{4}{36} = \frac{1}{9}$$ 5. **Probability that the difference of the two numbers is at most 1:** - Difference 0 means same number: 6 outcomes - Difference 1 means pairs like (1,2), (2,1), (2,3), (3,2), (3,4), (4,3), (4,5), (5,4), (5,6), (6,5): 10 outcomes - Total favorable outcomes: $$6 + 10 = 16$$ - Probability: $$\frac{16}{36} = \frac{4}{9}$$ 6. **Probability that the second number is a multiple of the first number:** - For each first number: - 1: multiples are all (1,1),(1,2),(1,3),(1,4),(1,5),(1,6) → 6 outcomes - 2: multiples are (2,2),(2,4),(2,6) → 3 outcomes - 3: multiples are (3,3),(3,6) → 2 outcomes - 4: multiples are (4,4) → 1 outcome - 5: multiples are (5,5) → 1 outcome - 6: multiples are (6,6) → 1 outcome - Total favorable outcomes: $$6 + 3 + 2 + 1 + 1 + 1 = 14$$ - Probability: $$\frac{14}{36} = \frac{7}{18}$$ **Final answers:** - Same number: $$\frac{1}{6}$$ - Odd prime on each roll: $$\frac{1}{9}$$ - Difference at most 1: $$\frac{4}{9}$$ - Second multiple of first: $$\frac{7}{18}$$