To find the probability of event $B$, where $B$ is defined as the sum of the two dice being at most 5, let's break it down:

Possible Outcomes

We need to consider all possible pairs of dice rolls and identify those where the sum is 5 or less.

List of Favorable Outcomes

Here are the combinations of dice rolls where the sum is at most 5:

• Sum = 2:

  • (1,1)

• Sum = 3:

  • (1,2)
  • (2,1)

• Sum = 4:

  • (1,3)
  • (2,2)
  • (3,1)

• Sum = 5:

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

These are the pairs of rolls where the sum of the two dice is 5 or less.

Counting Favorable Outcomes

Let's count the number of favorable outcomes:

• There is 1 way to get a sum of 2.
• There are 2 ways to get a sum of 3.
• There are 3 ways to get a sum of 4.
• There are 4 ways to get a sum of 5.

So, the total number of favorable outcomes is: $1 + 2 + 3 + 4 = 10$

Total Possible Outcomes

Since there are 6 faces on each die, the total number of possible outcomes when rolling two dice is: $6 \times 6 = 36$

Probability of Event B

The probability of event $B$ (the sum of the two dice being at most 5) is the ratio of the number of favorable outcomes to the total number of possible outcomes: $P(B) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{10}{36} = \frac{5}{18}$

Final Answer

The probability of event $B$ (the sum of the two dice being at most 5) is $\frac{5}{18}$.

Would you like more details on this, or do you have any further questions?

Here are five related questions to explore further:

1. What is the probability of the sum of the two dice being exactly 7?
2. What is the probability of the sum of the two dice being more than 8?
3. What is the probability of rolling doubles?
4. How does the probability change if we consider the sum being at most 4 instead of 5?
5. What is the probability of rolling a sum of 10 or higher?

Tip: When working with probabilities involving dice sums, it can be helpful to systematically list out all possible outcomes to avoid missing any combinations.