Let's walk through the problem displayed in the image.

Problem:

Two balanced dice are rolled, and all the possible outcomes are listed in the image.
Question: Determine the probability that the sum of the two dice is 3 or 12.
Answer choices:
A) $\frac{5}{36}$
B) $\frac{1}{12}$
C) $\frac{1}{18}$
D) $\frac{1}{9}$

Step-by-step solution:

1. Total number of outcomes:

When two six-sided dice are rolled, each die has 6 faces. So the total number of outcomes is:
$6 \times 6 = 36$
Each outcome in the table corresponds to a pair $(a, b)$ where $a$ and $b$ are the numbers showing on each die.

2. Identify favorable outcomes:

We are asked for the probability that the sum of the two dice is 3 or 12.

• Sum = 3:
  The pairs that give a sum of 3 are: $(1, 2), (2, 1)$
  So there are 2 favorable outcomes for this sum.

• Sum = 12:
  The pairs that give a sum of 12 are: $(6, 6)$
  So there is 1 favorable outcome for this sum.

• Total favorable outcomes:
  $2 + 1 = 3$

3. Calculate the probability:

The probability of an event is given by: $\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$ $\text{Probability} = \frac{3}{36} = \frac{1}{12}$

Final Answer:

The correct answer is B) $\frac{1}{12}$.

Do you have any questions or need further clarification?

Related Questions:

1. What is the probability of rolling a sum of 7 with two dice?
2. How many ways can the sum of two dice equal 8?
3. What is the probability of rolling doubles (both dice showing the same number)?
4. What is the complement of the event "the sum is 3 or 12"?
5. How does the probability change if the dice are weighted instead of balanced?

Tip: Always list all the favorable outcomes clearly when calculating probabilities to avoid missing any cases!