To find the probabilities for different outcomes when a pair of fair dice is tossed, we first need to consider the total possible outcomes. Since each die has 6 faces, the total number of outcomes is:

$6 \times 6 = 36$

(a) Probability of getting a total of 8

Next, we identify the pairs of dice that sum to 8:

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

There are 5 such pairs.

The probability of getting a total of 8 is the number of favorable outcomes divided by the total number of possible outcomes:

$P(\text{total} = 8) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{36}$

(b) Probability of getting at most a total of 5

Now, we identify the pairs of dice that sum to 5 or less:

• Total of 2: (1, 1)
• Total of 3: (1, 2), (2, 1)
• Total of 4: (1, 3), (2, 2), (3, 1)
• Total of 5: (1, 4), (2, 3), (3, 2), (4, 1)

There are 10 such pairs.

The probability of getting at most a total of 5 is:

$P(\text{total} \leq 5) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{10}{36} = \frac{5}{18}$

Summary

• Probability of getting a total of 8: $\frac{5}{36}$
• Probability of getting at most a total of 5: $\frac{5}{18}$

Would you like more details or have any questions?

Here are 8 related questions to expand your understanding:

1. What is the probability of rolling a total of 7 with a pair of dice?
2. How many outcomes result in a total greater than 8?
3. What is the probability of rolling doubles (e.g., (1,1), (2,2))?
4. What is the probability of getting a total of 2 or 12?
5. How does the probability change if one die is biased?
6. What is the expected value of the sum when rolling two dice?
7. How would you calculate the probability of rolling a total of 10?
8. How many outcomes result in a sum that is an odd number?

Tip: When working with probabilities, it's often useful to list all possible outcomes or create a probability distribution table to better visualize and understand the problem.