The question involves calculating probabilities for specific events concerning the roll of two six-sided dice. Here are the steps to compute the probabilities:

Event A: The sum is greater than 7.

1. Total possible outcomes when two dice are rolled = 6 * 6 = 36.
2. Favorable outcomes for sum > 7 (i.e., sums of 8, 9, 10, 11, 12):
   • Sum = 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes.
   • Sum = 9: (3,6), (4,5), (5,4), (6,3) → 4 outcomes.
   • Sum = 10: (4,6), (5,5), (6,4) → 3 outcomes.
   • Sum = 11: (5,6), (6,5) → 2 outcomes.
   • Sum = 12: (6,6) → 1 outcome.
   • Total favorable outcomes = 5 + 4 + 3 + 2 + 1 = 15.

Event B: The sum is not divisible by 5 and not divisible by 6.

1. We already know total possible outcomes = 36.
2. Sums from two dice that are divisible by 5 or 6:
   • Divisible by 5: 5, 10, 15, 20 → (5 is not possible, 15 and 20 are not possible with two dice, only 10 is valid: (4,6), (5,5), (6,4) → 3 outcomes).
   • Divisible by 6: 6, 12, 18 → (6: (1,5), (2,4), (3,3), (4,2), (5,1), 12: (6,6) → 7 outcomes).
   • Total for divisible by 5 or 6 = 3 + 7 = 10 outcomes.
   • Overlapping outcomes are only (6,4) and (4,6) from sum = 10, counted once in the divisible by 5 set.
   • Corrected count = 10 - 1 = 9.
   • Outcomes not divisible by either 5 or 6 = 36 - 9 = 27.

Now we can calculate the probabilities:

• Probability of Event A (P(A)) = Favorable outcomes for A / Total possible outcomes = 15 / 36.
• Probability of Event B (P(B)) = Outcomes not divisible by either 5 or 6 / Total possible outcomes = 27 / 36.

Let's compute these probabilities and round to two decimal places as requested.The probabilities for the events given are:

• $P(A)$ (The sum is greater than 7): 0.42
• $P(B)$ (The sum is not divisible by 5 and not divisible by 6): 0.75

Would you like to delve into any details or have any other questions on this?

Here are some related questions you might find interesting:

1. How would the probabilities change if the dice were not fair?
2. What is the probability of rolling a sum less than 7?
3. Can you calculate the probability of rolling an even number sum?
4. What are the chances of rolling a sum divisible by 3?
5. How do these probabilities translate to practical scenarios like games involving dice?

Tip:

Remember, in probability, knowing the total number of outcomes and how to categorize events based on those outcomes is crucial to determining the likelihood of any event.