Let's break down the problem:

Part (a): Probability that the sum of two 6-sided dice is 7

The total number of possible outcomes when rolling two dice is $6 \times 6 = 36$ since each die has 6 faces. Now, let's find the combinations where the sum is 7:

• (1,6)
• (2,5)
• (3,4)
• (4,3)
• (5,2)
• (6,1)

There are 6 combinations that give a sum of 7. Therefore, the probability of rolling a sum of 7 is: $P(\text{sum is } 7) = \frac{6}{36} = \frac{1}{6}$

Part (b): Probability that the sum is at most 7

Now, let's calculate how many outcomes have a sum of 7 or less. We'll list all the possible sums for each pair of dice:

• Sum = 2: (1,1) → 1 outcome
• Sum = 3: (1,2), (2,1) → 2 outcomes
• Sum = 4: (1,3), (2,2), (3,1) → 3 outcomes
• Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 outcomes
• Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes
• Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes

Summing all these, we get: $1 + 2 + 3 + 4 + 5 + 6 = 21 \text{ outcomes where the sum is at most 7}.$ So the probability is: $P(\text{sum is at most } 7) = \frac{21}{36} = \frac{7}{12}.$

Final Answers:

• (a) $P(\text{sum is 7}) = \frac{1}{6}$
• (b) $P(\text{sum is at most 7}) = \frac{7}{12}$

Do you want more details or have any questions?

Here are 5 related questions:

1. What is the probability that the sum is greater than 7?
2. What is the probability of rolling a double (both dice show the same number)?
3. What is the probability that the sum is 9?
4. What is the probability that the sum is 6?
5. What is the probability that the sum is at least 8?

Tip: In probability questions, always count the total possible outcomes first, then look for favorable outcomes to calculate probabilities.