2. Two dice are rolled.
(a) What is the conditional probability of at least one die showing a 5, given that the
sum is larger than 7?
(b) What is the conditional probability of the sum to be larger than 7, given that at
least one of the dice is showing a 5?

Let's tackle each part of the question step by step.

### Part (a): 
**What is the conditional probability of at least one die showing a 5, given that the sum is larger than 7?**

This is a conditional probability problem where we are asked to find:
\[
P(\text{at least one 5} \mid \text{sum} > 7)
\]
The formula for conditional probability is:
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]
Where:
- \( A \) = "At least one die shows a 5"
- \( B \) = "The sum is larger than 7"

First, we need to calculate \( P(B) \), i.e., the probability that the sum of two dice is greater than 7, and \( P(A \cap B) \), the probability that the sum is greater than 7 and at least one die shows a 5.

#### Total Sample Space:
The total sample space for rolling two dice is 36 (since each die has 6 faces, and \( 6 \times 6 = 36 \)).

#### Step 1: Calculate \( P(B) \) (sum > 7)
The possible sums greater than 7 are:
- 8: (2,6), (3,5), (4,4), (5,3), (6,2)
- 9: (3,6), (4,5), (5,4), (6,3)
- 10: (4,6), (5,5), (6,4)
- 11: (5,6), (6,5)
- 12: (6,6)

Counting the outcomes, there are 15 outcomes where the sum is greater than 7. So:
\[
P(B) = \frac{15}{36}
\]

#### Step 2: Calculate \( P(A \cap B) \) (at least one 5 and sum > 7)
Now, we need to count the outcomes where at least one die is a 5, and the sum is greater than 7. These outcomes are:
- (3,5), (4,5), (5,3), (5,4), (5,5), (5,6), (6,5)

There are 7 such outcomes. So:
\[
P(A \cap B) = \frac{7}{36}
\]

#### Step 3: Apply the conditional probability formula
Now, we can calculate \( P(A \mid B) \):
\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{7}{36}}{\frac{15}{36}} = \frac{7}{15}
\]

### Part (a) Answer:
The conditional probability that at least one die shows a 5, given that the sum is greater than 7, is:
\[
\boxed{\frac{7}{15}}
\]

---

### Part (b): 
**What is the conditional probability that the sum is larger than 7, given that at least one die is showing a 5?**

We need to find:
\[
P(\text{sum} > 7 \mid \text{at least one 5})
\]

Again, using the conditional probability formula:
\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)}
\]
Where:
- \( A \) = "At least one die shows a 5"
- \( B \) = "Sum is larger than 7"

#### Step 1: Calculate \( P(A) \) (at least one 5)
We need to count all outcomes where at least one die shows a 5. These outcomes are:
- (1,5), (2,5), (3,5), (4,5), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,5)

There are 11 such outcomes. So:
\[
P(A) = \frac{11}{36}
\]

#### Step 2: Recall \( P(A \cap B) \) (at least one 5 and sum > 7)
We already calculated this in part (a), and it is:
\[
P(A \cap B) = \frac{7}{36}
\]

#### Step 3: Apply the conditional probability formula
Now, we calculate \( P(B \mid A) \):
\[
P(B \mid A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{7}{36}}{\frac{11}{36}} = \frac{7}{11}
\]

### Part (b) Answer:
The conditional probability that the sum is larger than 7, given that at least one die is showing a 5, is:
\[
\boxed{\frac{7}{11}}
\]

---

### Do you want details or have any questions?

Here are some related questions to expand your understanding:

1. What is the probability of rolling a sum of exactly 7 with two dice?
2. What is the probability of getting doubles when rolling two dice?
3. What is the probability of getting at least one 6 when rolling two dice?
4. How would the problem change if one of the dice is weighted?
5. How do conditional probabilities apply to dependent and independent events?

**Tip:** Conditional probabilities are very useful when additional information helps refine the likelihood of an event occurring.

2. Two dice are rolled.
(a) What is the conditional probability of at least one die showing a 5, given that the sum is larger than 7?
(b) What is the conditional probability of the sum to be larger than 7, given that at least one of the dice is showing a 5?

Explore conditional probability using two dice rolls. Learn how to calculate the probability of getting at least one die showing a 5 given that the sum is greater than 7, and vice versa. Includes a detailed step-by-step solution, suitable for students in Grades 9-12.

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